From 5b9905b9f436de48aca4f84e1cbb3982cb4bece8 Mon Sep 17 00:00:00 2001 From: Ishani Gupta Date: Mon, 23 Oct 2023 16:18:44 -0700 Subject: [PATCH] minor fixes to prob 1 --- probability_1/probability_1.html | 1372 +++++++++++++++++ probability_1/probability_1.ipynb | 966 ++++++------ probability_1/probability_1.qmd | 72 +- .../libs/bootstrap/bootstrap.min.css | 4 +- 4 files changed, 1911 insertions(+), 503 deletions(-) create mode 100644 probability_1/probability_1.html diff --git a/probability_1/probability_1.html b/probability_1/probability_1.html new file mode 100644 index 00000000..f52fb274 --- /dev/null +++ b/probability_1/probability_1.html @@ -0,0 +1,1372 @@ + + + + + + + + + +Random Variables + + + + + + + + + + + + + + + + + + + + + +
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Random Variables

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  • Define a random variable in terms of its distribution
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  • Compute the expectation and variance of a random variable
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  • Gain familiarity with the Bernoulli and binomial random variables
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In the past few lectures, we’ve examined the role of complexity in influencing model performance. We’ve considered model complexity in the context of a tradeoff between two competing factors: model variance and training error.

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Thus far, our analysis has been mostly qualitative. We’ve acknowledged that our choice of model complexity needs to strike a balance between model variance and training error, but we haven’t yet discussed why exactly this tradeoff exists.

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To better understand the origin of this tradeoff, we will need to introduce the language of random variables. The next two lectures on probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called bias-variance tradeoff. Our roadmap for the next few lectures will be:

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  1. Random Variables Estimators: introduce random variables, considering the concepts of expectation, variance, and covariance
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  2. Estimators, Bias, and Variance: re-express the ideas of model variance and training error in terms of random variables and use this new perspective to investigate our choice of model complexity
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+Data 8 +
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Recall the following concepts from Data 8:

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  1. Sample mean: the mean of your random sample

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  2. Central Limit Theorem: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean

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    1. is roughly normal

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    2. is centered at the population mean

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    3. has an \(SD = \frac{\text{population SD}}{\sqrt{\text{sample size}}}\)

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Random Variables and Distributions

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Suppose we generate a set of random data, like a random sample from some population. A random variable is a numerical function of the randomness in the data. It is random since our sample was drawn at random; it is variable because its exact value depends on how this random sample came out. As such, the domain or input of our random variable is all possible (random) outcomes in a sample space, and its range or output is the number line. We typically denote random variables with uppercase letters, such as \(X\) or \(Y\).

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Distribution

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For any random variable \(X\), we need to be able to specify 2 things: 1. Possible values: the set of values the random variable can take on. 2. Probabilities: the set of probabilities describing how the total probability of 100% is split over the possible values.

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If \(X\) is discrete (has a finite number of possible values), * The probability that a random variable \(X\) takes on the value \(x\) is given by \(P(X=x)\). * Probabilities must sum to 1: \(\sum_{\text{all} x} P(X=x) = 1\),

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We can often display this using a probability distribution table (example shown below).

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The distribution of a random variable \(X\) is a description of how the total probability of 100% is split over all the possible values of \(X\), and it fully defines a random variable.

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The distribution of a discrete random variable can also be represented using a histogram. If a variable is continuous – it can take on infinitely many values – we can illustrate its distribution using a density curve.

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Probabilities are areas. For discrete random variables, the area of the red bars represent the probability that a discrete random variable \(X\) falls within those values. For continuous random variables, the area under the curve represents the probability that a discrete random variable \(Y\) falls within those values.

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If we sum up the total area of the bars/under the density curve, we should get 100%, or 1.

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Example: Tossing a Coin

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To give a concrete example, let’s formally define a fair coin toss. A fair coin can land on heads (\(H\)) or tails (\(T\)), each with a probability of 0.5. With these possible outcomes, we can define a random variable \(X\) as: \[X = \begin{cases} + 1, \text{if the coin lands heads} \\ + 0, \text{if the coin lands tails} + \end{cases}\]

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\(X\) is a function with a domain (input) of \(\{H, T\}\) and a range (output) of \(\{1, 0\}\). We can write this in function notation as \[\begin{cases} X(H) = 1 \\ X(T) = 0 \end{cases}\] and the probability distribution table of \(X\) is

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\(x\)\(P(X=x)\)
0\(\frac{1}{2}\)
1\(\frac{1}{2}\)
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Suppose we draw a random sample \(s\) of size 3 from all students enrolled in Data 100. We can define \(Y\) as the number of data science students in our sample. Its domain is all possible samples of size 3, and its range is \(\{0, 1, 2, 3\}\).

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We can show the distribution of \(Y\) in the following tables. The table on the left lists all possible samples of \(s\) and the number of times they can appear (\(Y(s)\)). We can use this to calculate the values for the table on the right, a probability distribution table.

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Simulation

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Given a random variable \(X\)’s distribution, how could we generate/simulate a population? To do so, we can randomly pick values of \(X\) according to its distribution using np.random.choice or df.sample.

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Expectation and Variance

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There are several ways to describe a random variable. The methods shown above - table of all samples \(s, X(s)\), distribution table \(P(X=x)\), and histograms - are all definitions that fully describe a random variable. Often, it is easier to describe a random variable using some numerical summary, rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a “summary” of how the variable tends to behave, they are not random – think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable.

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Expectation

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The expectation of a random variable \(X\) is the weighted average of the values of \(X\), where the weights are the probabilities of each value occurring. There are two equivalent ways to compute the expectation: 1. Apply the weights one sample at a time: \[\mathbb{E}[X] = \sum_{\text{all possible } s} X(s) P(s)\]. 2. Apply the weights one possible value at a time: \[\mathbb{E}[X] = \sum_{\text{all possible } x} x P(X=x)\]

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We want to emphasize that the expectation is a number, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram. If we simulate the variable many times, it is the long-run average of the random variable.

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Example 1: Coin Toss

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Going back to our coin toss example, we define a random variable \(X\) as: \[X = \begin{cases} + 1, \text{if the coin lands heads} \\ + 0, \text{if the coin lands tails} + \end{cases}\] We can calculate its expectation \(\mathbb{E}[X]\) using the second method of applying the weights one possible value at a time: \[\begin{align} +\mathbb{E}[X] &= \sum_{x} x P(X=x) \\ +&= 1 * 0.5 + 0 * 0.5 \\ +&= 0.5 +\end{align}\] Note that \(\mathbb{E}[X] = 0.5\) is not a possible value of \(X\); it’s an average. The expectation of X does not need to be a possible value of X.

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Example 2

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Consider the random variable \(X\):

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\(x\)\(P(X=x)\)
30.1
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60.4
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To calculate it’s expectation, \[\begin{align} +\mathbb{E}[X] &= \sum_{x} x P(X=x) \\ +&= 3 * 0.1 + 4 * 0.2 + 6 * 0.4 + 8 * 0.3 \\ +&= 0.3 + 0.8 + 2.4 + 2.4 \\ +&= 5.9 +\end{align}\] Again, note that \(\mathbb{E}[X] = 5.9\) is not a possible value of \(X\); it’s an average. The expectation of X does not need to be a possible value of X.

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Variance

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The variance of a random variable is a measure of its chance error. It is defined as the expected squared deviation from the expectation of \(X\). Put more simply, variance asks: how far does \(X\) typically vary from its average value, just by chance? What is the spread of \(X\)’s distribution?

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\[\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2]\]

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The units of variance are the square of the units of \(X\). To get it back to the right scale, use the standard deviation of \(X\): \(\text{SD}(X) = \sqrt{\text{Var}(X)}\).

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Like with expectation, variance is a number, not a random variable! Its main use is to quantify chance error.

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By Chebyshev’s inequality, which you saw in Data 8, no matter what the shape of the distribution of X is, the vast majority of the probability lies in the interval “expectation plus or minus a few SDs.”

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If we expand the square and use properties of expectation, we can re-express variance as the computational formula for variance. This form is often more convenient to use when computing the variance of a variable by hand, and it is also useful in Mean Squared Error calculations (if \(X\) is centered and \(E(X)=0\), then \(\mathbb{E}[X^2] = \text{Var}(X)\)).

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\[\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\]

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\[\begin{align} + \text{Var}(X) &= \mathbb{E}[(X-\mathbb{E}[X])^2] \\ + &= \mathbb{E}(X^2 - 2X\mathbb{E}(X) + (\mathbb{E}(X))^2) \\ + &= \mathbb{E}(X^2) - 2 \mathbb{E}(X)\mathbb{E}(X) +( \mathbb{E}(X))^2\\ + &= \mathbb{E}[X^2] - (\mathbb{E}[X])^2 + \end{align}\]

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How do we compute \(\mathbb{E}[X^2]\)? Any function of a random variable is also a random variable – that means that by squaring \(X\), we’ve created a new random variable. To compute \(\mathbb{E}[X^2]\), we can simply apply our definition of expectation to the random variable \(X^2\).

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\[\mathbb{E}[X^2] = \sum_{x} x^2 P(X = x)\]

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Example: Dice

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Let \(X\) be the outcome of a single fair dice roll. \(X\) is a random variable defined as \[X = \begin{cases} + \frac{1}{6}, \text{if } x \in \{1,2,3,4,5,6\} \\ + 0, \text{otherwise} + \end{cases}\]

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\[ \begin{align} + \mathbb{E}[X] &= 1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6}) \\ + &= (\frac{1}{6}) ( 1 + 2 + 3 + 4 + 5 + 6) \\ + &= \frac{7}{2} + \end{align}\]

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Using approach 1: \[\begin{align} + \text{Var}(X) &= (\frac{1}{6})((1 - \frac{7}{2})^2 + (2 - \frac{7}{2})^2 + (3 - \frac{7}{2})^2 + (4 - \frac{7}{2})^2 + (5 - \frac{7}{2})^2 + (6 - \frac{7}{2})^2) \\ + &= \frac{35}{12} + \end{align}\]

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Using approach 2: \[\mathbb{E}[X^2] = \sum_{x} x^2 P(X = x) = \frac{91}{6}\] \[\text{Var}(X) = \frac{91}{6} - (\frac{7}{2})^2 = \frac{35}{12}\]

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Sums of Random Variables

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Often, we will work with multiple random variables at the same time. A function of a random variable is also a random variable, so if you create multiple random variables based on your sample, then functions of those random variables are also random variables.

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For example, if \(X_1, X_2, ..., X_n\) are random variables, then so are all of these: * \(X_n^2\) * \(\#\{i : X_i > 10\}\) * \(\text{max}(X_1, X_2, ..., X_n)\) * \(\frac{1}{n} \sum_{i=1}^n (X_i - c)^2\) * \(\frac{1}{n} \sum_{i=1}^n X_i\)

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Equal vs. Identically Distributed vs. i.i.d

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Suppose that we have two random variables \(X\) and \(Y\):

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  • \(X\) and \(Y\) are equal if \(X(s) = Y(s)\) for every sample \(s\). Regardless of the exact sample drawn, \(X\) is always equal to \(Y\).
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  • \(X\) and \(Y\) are identically distributed if the distribution of \(X\) is equal to the distribution of \(Y\). We say “X and Y are equal in distribution.” That is, \(X\) and \(Y\) take on the same set of possible values, and each of these possible values is taken with the same probability. On any specific sample \(s\), identically distributed variables do not necessarily share the same value. If X = Y, then X and Y are identically distributed; however, the converse is not true (ex: Y = 7-X, X is a die)
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  • \(X\) and \(Y\) are independent and identically distributed (i.i.d) if +
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    1. The variables are identically distributed.
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    2. Knowing the outcome of one variable does not influence our belief of the outcome of the other.
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For example, let \(X_1\) and \(X_2\) be numbers on rolls of two fair die. \(X_1\) and \(X_2\) are i.i.d, so \(X_1\) and \(X_2\) have the same distribution. However, the sums \(Y = X_1 + X_1 = 2X_1\) and \(Z=X_1+X_2\) have different distributions but the same expectation.

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However, \(Y = X_1\) has a larger variance

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Properties of Expectation

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Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: \[\mathbb{E}[X] = \sum_{x} x P(X=x)\].

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1.Linearity of expectation. The expectation of the linear transformation \(aX+b\), where \(a\) and \(b\) are constants, is:

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\[\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\]

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\[\begin{align} + \mathbb{E}[aX+b] &= \sum_{x} (ax + b) P(X=x) \\ + &= \sum_{x} (ax P(X=x) + bP(X=x)) \\ + &= a\sum_{x}P(X=x) + b\sum_{x}P(X=x)\\ + &= a\mathbb{E}(X) = b * 1 + \end{align}\]

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  1. Expectation is also linear in sums of random variables.
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\[\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]\]

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\[\begin{align} + \mathbb{E}[X+Y] &= \sum_{s} (X+Y)(s) P(s) \\ + &= \sum_{s} (X(s)P(s) + Y(s)P(s)) \\ + &= \sum_{s} X(s)P(s) + \sum_{s} Y(s)P(s)\\ + &= \mathbb{E}[X] + \mathbb{E}[Y] +\end{align}\]

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  1. If \(g\) is a non-linear function, then in general, \[\mathbb{E}[g(X)] \neq g(\mathbb{E}[X])\].
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  • For example, if \(X\) is -1 or 1 with equal probability, then \(\mathbb{E}[X] = 0\) but \(\mathbb{E}[X^2] = 1 \neq 0\)
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Properties of Variance

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Recall the definition of variance: \[\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2]\].

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  1. Unlike expectation, variance is non-linear. The variance of the linear transformation \(aX+b\) is: \[\text{Var}(aX+b) = a^2 \text{Var}(X)\].
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  • The full proof of this fact can be found using the definition of variance. As general intuition, consider that \(aX+b\) scales the variable \(X\) by a factor of \(a\), then shifts the distribution of \(X\) by \(b\) units.
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We know that \[\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\].

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In order to compute \(\text{Var}(aX+b)\), consider that a shift by b units does not affect spread, so \(\text{Var}(aX+b) = \text{Var}(aX)\).

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Then, \[\begin{align} + \text{Var}(aX+b) &= \text{Var}(aX) \\ + &= E((aX)^2) - (E(aX))^2 + &= E(a^2 X^2) - (aE(X))^2\\ + &= a^2 (E(X^2) - (E(X))^2) \\ + &= a^2 \text{Var}(X) +\end{align}\]

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  • Shifting the distribution by \(b\) does not impact the spread of the distribution. Thus, \(\text{Var}(aX+b) = \text{Var}(aX)\).
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  • Scaling the distribution by \(a\) does impact the spread of the distribution.
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  1. Variance of sums of RVs is affected by the (in)dependence of the RVs. \[\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) 2\text{cov}(X,Y)\] \[\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \qquad \text{if } X, Y \text{ independent}\]
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The variance of a sum is affected by the dependence between the two random variables that are being added. Let’s expand out the definition of \(\text{Var}(X + Y)\) to see what’s going on.

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To simplify the math, let \(\mu_x = \mathbb{E}[X]\) and \(\mu_y = \mathbb{E}[Y]\).

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\[ \begin{align} +\text{Var}(X + Y) &= \mathbb{E}[(X+Y- \mathbb{E}(X+Y))^2] \\ +&= \mathbb{E}[((X - \mu_x) + (Y - \mu_y))^2] \\ +&= \mathbb{E}[(X - \mu_x)^2 + 2(X - \mu_x)(Y - \mu_y) + (Y - \mu_y)^2] \\ +&= \mathbb{E}[(X - \mu_x)^2] + \mathbb{E}[(Y - \mu_y)^2] + \mathbb{E}[(X - \mu_x)(Y - \mu_y)] \\ +&= \text{Var}(X) + \text{Var}(Y) + \mathbb{E}[(X - \mu_x)(Y - \mu_y)] +\end{align}\]

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Covariance and Correlation

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We define the covariance of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to two random variables: \(\text{Cov}(X, X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \text{Var}(X)\).

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\[\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]\]

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We can treat the covariance as a measure of association. Remember the definition of correlation given when we first established SLR?

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\[r(X, Y) = \mathbb{E}\left[\left(\frac{X-\mathbb{E}[X]}{\text{SD}(X)}\right)\left(\frac{Y-\mathbb{E}[Y]}{\text{SD}(Y)}\right)\right] = \frac{\text{Cov}(X, Y)}{\text{SD}(X)\text{SD}(Y)}\]

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It turns out we’ve been quietly using covariance for some time now! If \(X\) and \(Y\) are independent, then \(\text{Cov}(X, Y) =0\) and \(r(X, Y) = 0\). Note, however, that the converse is not always true: \(X\) and \(Y\) could have \(\text{Cov}(X, Y) = r(X, Y) = 0\) but not be independent.

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Summary

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  • Let \(X\) be a random variable with distribution $P(X=x). +
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    • \(\mathbb{E}[X] = \sum_{x} x P(X=x)\)
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    • \(\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\)
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  • Let \(a\) and \(b\) be scalar values. +
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    • \(\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\)
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    • \(\text{Var}(aX+b) = a^2 \text{Var}(X)\)
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  • Let \(Y\) be another random variable. +
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    • \(\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]\)
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    • \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) 2\text{cov}(X,Y)\)
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Common Random Variables

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There are several cases of random variables that appear often and have useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable’s shape (i.e., distribution) and its values.

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    • Let X be a Bernoulli(p) random variable +
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      • \(\mathbb{E}[X] = 1 * p + 0 * (1-p) = p\) +
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        • \(\mathbb{E}[X^2] = 1^2 * p + 0 * (1-p) = p\)
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      • \(\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = p - p^2 = p(1-p)\)
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  • Binomial(n, p) +
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    • Let \(Y\) be a Binomial(n, p) random variable. +
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      • The distribution of \(Y\) is given by the binomial formula, and we can write \(Y = \sum_{i=1}^n X_i\) where: +
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        • \(X_i\) s the indicator of success on trial i. \(X_i = 1\) if trial i is a success, else 0.
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        • All \(X_i\) are i.i.d. and Bernoulli(p).
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      • \(\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[X_i] = np\)
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      • \(\text{Var}(X) = \sum_{i=1}^n \text{Var}(X_i) = np(1-p)\) +
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        • \(X_i\)’s are independent, so \(\text{Cov}(X_i, X_j) = 0\) for all i, j.
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  • Uniform on a finite set of values +
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    • Probability of each value is 1 / (number of possible values).
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    • For example, a standard/fair die.
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  • Uniform on the unit interval (0, 1) +
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    • Density is flat at 1 on (0, 1) and 0 elsewhere.
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  • Normal(\(\mu, \sigma^2\)) +
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    • \[f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{\!2}\,\right)\]
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Populations and Samples

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Today, we’ve talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, variance, functions of the random variable, etc.

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In Data Science, however, we often do not have access to the whole population, so we don’t know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer properties of the population.

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When sampling, we make the (big) assumption that we sample uniformly at random with replacement from the population; each observation in our sample is a random variable drawn i.i.d from our population distribution.

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Sample Mean

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Consider an i.i.d. sample \(X_1, X_2, ..., X_n\) drawn from a population with mean 𝜇 and SD 𝜎. We define the sample mean as \[\bar{X_n} = \frac{1}{n} \sum_{i=1}^n X_i\]

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The expectation of the sample mean is given by: \[\begin{align} + \mathbb{E}[\bar{X_n}] &= \frac{1}{n} \sum_{i=1}^n \mathbb{E}[X_i] \\ + &= \frac{1}{n} (n \mu) \\ + &= \mu +\end{align}\]

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The variance is given by: \[\begin{align} + \text{Var}(\bar{X_n}) &= \frac{1}{n^2} \text{Var}( \sum_{i=1}^n X_i) \\ + &= \frac{1}{n^2} \left( \sum_{i=1}^n \text{Var}(X_i) \right) \\ + &= \frac{1}{n^2} (n \sigma^2) = \frac{\sigma^2}{n} +\end{align}\]

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\(\bar{X_n}\) is normally distributed by the Central Limit Theorem (CLT).

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Central Limit Theorem

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The CLT states that no matter what population you are drawing from, if an i.i.d. sample of size \(n\) is large, the probability distribution of the sample mean is roughly normal with mean 𝜇 and SD \(\sigma/\sqrt{n}\).

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Any theorem that provides the rough distribution of a statistic and doesn’t need the distribution of the population is valuable to data scientists because we rarely know a lot about the population!

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For a more in-depth demo check out onlinestatbook.

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The CLT applies if the sample size \(n\) is large, but how large does n have to be for the normal approximation to be good? It depends on the shape of the distribution of the population.

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  • If the population is roughly symmetric and unimodal/uniform, could need as few as \(n = 20\).
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  • If the population is very skewed, you will need a bigger n.
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  • If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped.
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Using the Sample Mean to Estimate the Population Mean

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Our goal with sampling is often to estimate some characteristic of a population. When we collect a single sample, it has just one average. Since our sample was random, it could have come out differently. The CLT helps us understand this difference. We should consider the average value and spread of all possible sample means, and what this means for how big \(n\) should be.

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For every sample size, the expected value of the sample mean is the population mean. \(\mathbb{E}[\bar{X_n}] = \mu\). We call the sample mean an unbiased estimator of the population mean, and we’ll cover this more in the next lecture.

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The square root law (Data 8) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. \(\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}\). The sample mean is more likely to be close to the population mean if we have a larger sample size.

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+ + + + \ No newline at end of file diff --git a/probability_1/probability_1.ipynb b/probability_1/probability_1.ipynb index 7f7e1cce..d0eb884c 100644 --- a/probability_1/probability_1.ipynb +++ b/probability_1/probability_1.ipynb @@ -1,474 +1,498 @@ { - "cells": [ - { - "cell_type": "raw", - "metadata": {}, - "source": [ - "---\n", - "title: Random Variables\n", - "execute:\n", - " echo: true\n", - "format:\n", - " html:\n", - " code-fold: true\n", - " code-tools: true\n", - " toc: true\n", - " toc-title: Random Variables\n", - " page-layout: full\n", - " theme:\n", - " - cosmo\n", - " - cerulean\n", - " callout-icon: false\n", - "---" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "::: {.callout-note collapse=\"true\"}\n", - "## Learning Outcomes\n", - "* Define a random variable in terms of its distribution\n", - "* Compute the expectation and variance of a random variable\n", - "* Gain familiarity with the Bernoulli and binomial random variables\n", - ":::\n", - "\n", - "In the past few lectures, we've examined the role of complexity in influencing model performance. We've considered model complexity in the context of a tradeoff between two competing factors: model variance and training error. \n", - "\n", - "Thus far, our analysis has been mostly qualitative. We've acknowledged that our choice of model complexity needs to strike a balance between model variance and training error, but we haven't yet discussed *why* exactly this tradeoff exists.\n", - "\n", - "To better understand the origin of this tradeoff, we will need to introduce the language of **random variables**. The next two lectures of probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called **bias-variance tradeoff**. Our roadmap for the next few lectures will be:\n", - "\n", - "1. Random Variables Estimators: introduce random variables, considering the concepts of expectation, variance, and covariance\n", - "2. Estimators, Bias, and Variance: re-express the ideas of model variance and training error in terms of random variables and use this new perspective to investigate our choice of model complexity\n", - "\n", - "::: {.callout-tip}\n", - "## Data 8\n", - "Recall the following concepts from Data 8: \n", - "\n", - "1. sample mean: the mean of your random sample\n", - "2. Central Limit Theorem: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean\n", - "\n", - " a. is roughly normal\n", - "\n", - " b. is centered at the population mean\n", - " \n", - " c. has an $SD = \\frac{\\text{population SD}}{\\sqrt{\\text{sample size}}}$\n", - "\n", - ":::" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random Variables and Distributions\n", - "\n", - "Suppose we generate a set of random data, like a random sample from some population. A **random variable** is a *numerical function* of the randomness in the data. It is *random* since our sample was drawn at random; it is *variable* because its exact value depends on how this random sample came out. As such, the domain or input of our random variable is all possible (random) outcomes in a *sample space*, and its range or output is the number line. We typically denote random variables with uppercase letters, such as $X$ or $Y$. \n", - "\n", - "### Distribution\n", - "For any random variable $X$, we need to be able to specify 2 things: \n", - "1. possible values: the set of values the random variable can take on\n", - "2. probabilities: the set of probabilities describing how the total probability of 100% is split over the possible values.\n", - "\n", - "If $X$ is discrete (has a finite number of possible values),\n", - "* The probability that a random variable $X$ takes on the value $x$ is given by $P(X=x)$.\n", - "* Probabilities must sum to 1: $\\sum_{\\text{all} x} P(X=x) = 1$\n", - "* we can often do this using a **probability distribution table** (example shown below).\n", - "\n", - "The **distribution** of a random variable $X$ is a description of how the total probability of 100% is split over all the possible values of $X$, and it fully defines a random variable.\n", - "\n", - "The distribution of a discrete random variable can also be represented using a histogram. If a variable is **continuous** – it can take on infinitely many values – we can illustrate its distribution using a density curve. \n", - "

\n", - "discrete_continuous\n", - "

\n", - "Probabilities are areas. For discrete random variables, the *area of the red bars* represent the probability that a discrete random variable $X$ falls within those values. For continuous random variables, the *area under the curve* represents the probability that a discrete random variable $Y$ falls within those values.\n", - "

\n", - "discrete_continuous\n", - "

\n", - "If we sum up the total area of the bars/under the density curve, we should get 100%, or 1.\n", - "\n", - "\n", - "### Example: Tossing a Coin\n", - "To give a concrete example, let's formally define a fair coin toss. A fair coin can land on heads ($H$) or Tails ($T$), each with a probability of 0.5. With these possible outcomes, we can define a random variable $X$ as: \n", - "$$X = \\begin{cases} \n", - " 1, \\text{if the coin lands heads} \\\\\n", - " 0, \\text{if the coin lands tails} \n", - " \\end{cases}$$\n", - "\n", - "$X$ is a function with a domain (input) of $\\{H, T\\}$ and a range (output) of $\\{1, 0\\}$. We can write this in function notation as \n", - "$$\\begin{cases} X(H) = 1 \\\\ X(T) = 0 \\end{cases}$$\n", - "and the probability distribution table of $X$ is \n", - "\n", - "| $x$ | $P(X=x)$ | \n", - "| --- | -------- |\n", - "| 0 | $\\frac{1}{2}$ | \n", - "| 1 | $\\frac{1}{2}$ |\n", - "\n", - "Suppose we draw a random sample $s$ of size 3 from all students enrolled in Data 100. \n", - "We can define $Y$ as the number of data science students in our sample. It's domain is all possible samples of size 3, and its range is $\\{0, 1, 2, 3\\}$.\n", - "\n", - "

\n", - " rv\n", - "

\n", - "\n", - "We can show the distribution of $Y$ in the following tables. The table on the left lists all possible samples of $s$ and the number of times they can appear ($Y(s)$). We can use this to calculate the values for the table on the right, a **probability distribution table**. \n", - "

\n", - "distribution\n", - "

\n", - "\n", - "### Simulation\n", - "Given a random variable $X$’s distribution, how could we **generate/simulate** a population? To do so, we can randomly pick values of $X$ according to its distribution using `np.random.choice` or `df.sample`. " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Expectation and Variance\n", - "There are several ways to describe a random variable. The methods shown above - table of all samples $s, X(s)$, distribution table $P(X=x)$, histograms - are all definitions that fully describe a random variable. Often, it is easier to describe a random variable using some numerical summary, rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a \"summary\" of how the variable tends to behave, they are *not* random – think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable.\n", - "\n", - "### Expectation\n", - "The **expectation** of a random variable $X$ is the weighted average of the values of $X$, where the weights are the probabilities of each value occurring. There are two equivalent ways to compute the expectation: \n", - "1. Apply the weights one *sample* at a time: $$\\mathbb{E}[X] = \\sum_{\\text{all possible } s} X(s) P(s)$$.\n", - "2. Apply the weights one possible *value* at a time: $$\\mathbb{E}[X] = \\sum_{\\text{all possible } x} x P(X=x)$$\n", - "\n", - "We want to emphasize that the expectation is a *number*, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram. If we simulate the variable many times, it is the long run average of the random variable.\n", - "\n", - "### Example 1: Coin Toss\n", - "Going back to our coin toss example, we define a random variable $X$ as: \n", - "$$X = \\begin{cases} \n", - " 1, \\text{if the coin lands heads} \\\\\n", - " 0, \\text{if the coin lands tails} \n", - " \\end{cases}$$\n", - "We can calculate the its expectation $\\mathbb{E}[X]$ using the second method of applying the weights one possible value at a time: \n", - "$$\\begin{align}\n", - " \\mathbb{E}[X] &= \\sum_{x} x P(X=x) \\\\\n", - " &= 1 * 0.5 + 0 * 0.5 \\\\\n", - " &= 0.5\n", - "\\end{align}$$\n", - "Note that $\\mathbb{E}[X] = 0.5$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**.\n", - "\n", - "### Example 2\n", - "Consider the random variable $X$: \n", - "\n", - "| $x$ | $P(X=x)$ | \n", - "| --- | -------- |\n", - "| 3 | 0.1 | \n", - "| 4 | 0.2 |\n", - "| 6 | 0.4 | \n", - "| 8 | 0.3 |\n", - "\n", - "To calculate it's expectation, \n", - "$$\\begin{align}\n", - " \\mathbb{E}[X] &= \\sum_{x} x P(X=x) \\\\\n", - " &= 3 * 0.1 + 4 * 0.2 + 6 * 0.4 + 8 * 0.3 \\\\\n", - " &= 0.3 + 0.8 + 2.4 + 2.4 \\\\\n", - " &= 5.9\n", - "\\end{align}$$\n", - "Again, note that $\\mathbb{E}[X] = 5.9$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**.\n", - "\n", - "### Variance\n", - "The **variance** of a random variable is a measure of its chance error. It is defined as the expected squared deviation from the expectation of $X$. Put more simply, variance asks: how far does $X$ typically vary from its average value, just by chance? What is the spread of $X$'s distribution?\n", - "\n", - "$$\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2]$$\n", - "\n", - "The units of variance are the square of the units of $X$. To get it back to the right scale, use the standard deviation of $X$: $\\text{SD}(X) = \\sqrt{\\text{Var}(X)}$.\n", - "\n", - "Like with expectation, **variance is a number, not a random variable**! It's main use is to quantify chance error.\n", - "\n", - "By [Chebyshev’s inequality](https://www.inferentialthinking.com/chapters/14/2/Variability.html#Chebychev's-Bounds), which you saw in Data 8, no matter what the shape of the distribution of X is, the vast majority of the probability lies in the interval “expectation plus or minus a few SDs.”\n", - "\n", - "If we expand the square and use properties of expectation, we can re-express variance as the **computational formula for variance**. This form is often more convenient to use when computing the variance of a variable by hand, and it is also useful in Mean Squared Error calculations (if $X$ is centered and $E(X)=0$, then $\\mathbb{E}[X^2] = \\text{Var}(X)$).\n", - "\n", - "$$\\text{Var}(X) = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2$$\n", - "\n", - "
\n", - " Proof (toggle this cell)\n", - " \n", - " $$\\begin{align}\n", - " \\text{Var}(X) &= \\mathbb{E}[(X-\\mathbb{E}[X])^2] \\\\\n", - " &= \\mathbb{E}(X^2 - 2X\\mathbb{E}(X) + (\\mathbb{E}(X))^2) \\\\\n", - " &= \\mathbb{E}(X^2) - 2 \\mathbb{E}(X)\\mathbb{E}(X) +( \\mathbb{E}(X))^2\\\\\n", - " &= \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2\n", - " \\end{align}$$\n", - "
\n", - "\n", - "How do we compute $\\mathbb{E}[X^2]$? Any function of a random variable is *also* a random variable – that means that by squaring $X$, we've created a new random variable. To compute $\\mathbb{E}[X^2]$, we can simply apply our definition of expectation to the random variable $X^2$.\n", - "\n", - "$$\\mathbb{E}[X^2] = \\sum_{x} x^2 P(X = x)$$ \n", - "\n", - "### Example: Dice\n", - "Let $X$ be the outcome of a single fair dice roll. $X$ is a random variable defined as \n", - "$$X = \\begin{cases} \n", - " \\frac{1}{6}, \\text{if } x \\in \\{1,2,3,4,5,6\\} \\\\\n", - " 0, \\text{otherwise} \n", - " \\end{cases}$$\n", - "\n", - "::: {.callout-caution collapse=\"true\"}\n", - "## What's the expectation $\\mathbb{E}[X]?$\n", - "\n", - "$$ \\begin{align} \n", - " \\mathbb{E}[X] &= 1(\\frac{1}{6}) + 2(\\frac{1}{6}) + 3(\\frac{1}{6}) + 4(\\frac{1}{6}) + 5(\\frac{1}{6}) + 6(\\frac{1}{6}) \\\\\n", - " &= (\\frac{1}{6}) ( 1 + 2 + 3 + 4 + 5 + 6) \\\\\n", - " &= \\frac{7}{2}\n", - " \\end{align}$$\n", - ":::\n", - "\n", - "::: {.callout-caution collapse=\"true\"}\n", - "## What's the variance $\\text{Var}(X)?$\n", - "\n", - "Using approach 1: \n", - " $$\\begin{align} \n", - " \\text{Var}(X) &= (\\frac{1}{6})((1 - \\frac{7}{2})^2 + (2 - \\frac{7}{2})^2 + (3 - \\frac{7}{2})^2 + (4 - \\frac{7}{2})^2 + (5 - \\frac{7}{2})^2 + (6 - \\frac{7}{2})^2) \\\\\n", - " &= \\frac{35}{12}\n", - " \\end{align}$$\n", - "\n", - "Using approach 2: \n", - "$$\\mathbb{E}[X^2] = \\sum_{x} x^2 P(X = x) = \\frac{91}{6}$$\n", - "$$\\text{Var}(X) = \\frac{91}{6} - (\\frac{7}{2})^2 = \\frac{35}{12}$$\n", - ":::" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Sums of Random Variables\n", - "Often, we will work with multiple random variables at the same time. A function of a random variable is also a random variable, so if you create multiple random variables based on your sample, then functions of those random variables are also random variables.\n", - "\n", - "For example, if $X_1, X_2, ..., X_n$ are random variables, then so are all of these: \n", - "* $X_n^2$\n", - "* $\\#\\{i : X_i > 10\\}$\n", - "* $\\text{max}(X_1, X_2, ..., X_n)$\n", - "* $\\frac{1}{n} \\sum_{i=1}^n (X_i - c)^2$\n", - "* $\\frac{1}{n} \\sum_{i=1}^n X_i$\n", - "\n", - "\n", - "### Equal vs. Identically Distributed vs. i.i.d\n", - "Suppose that we have two random variables $X$ and $Y$:\n", - "\n", - "* $X$ and $Y$ are **equal** if $X(s) = Y(s)$ for every sample $s$. Regardless of the exact sample drawn, $X$ is always equal to $Y$.\n", - "* $X$ and $Y$ are **identically distributed** if the distribution of $X$ is equal to the distribution of $Y$. We say “X and Y are equal in distribution.” That is, $X$ and $Y$ take on the same set of possible values, and each of these possible values is taken with the same probability. On any specific sample $s$, identically distributed variables do *not* necessarily share the same value. If X = Y, then X and Y are identically distributed; however, the converse is not true (ex: Y = 7-X, X is a die)\n", - "* $X$ and $Y$ are **independent and identically distributed (i.i.d)** if \n", - " 1. the variables are identically distributed and\n", - " 2. knowing the outcome of one variable does not influence our belief of the outcome of the other.\n", - "\n", - "For example, let $X_1$ and $X_2$ be numbers on rolls of two fair die. $X_1$ and $X_2$ are i.i.d, so $X_1$ and $X_2$ have the same distribution. However, the sums $Y = X_1 + X_1 = 2X_1$ and $Z=X_1+X_2$ have different distributions but the same expectation.\n", - "

\n", - "distribution\n", - "

\n", - "However, $Y = X_1$ has a larger variance\n", - "

\n", - "distribution\n", - "

\n", - "\n", - "### Properties of Expectation \n", - "Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: $$\\mathbb{E}[X] = \\sum_{x} x P(X=x)$$\n", - "\n", - "1.**Linearity of expectation**. The expectation of the linear transformation $aX+b$, where $a$ and $b$ are constants, is:\n", - "\n", - "$$\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$$\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "$$\\begin{align}\n", - " \\mathbb{E}[aX+b] &= \\sum_{x} (ax + b) P(X=x) \\\\\n", - " &= \\sum_{x} (ax P(X=x) + bP(X=x)) \\\\\n", - " &= a\\sum_{x}P(X=x) + b\\sum_{x}P(X=x)\\\\\n", - " &= a\\mathbb{E}(X) = b * 1\n", - " \\end{align}$$\n", - ":::\n", - "\n", - "2. Expectation is also linear in *sums* of random variables. \n", - "\n", - "$$\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y]$$\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "$$\\begin{align}\n", - " \\mathbb{E}[X+Y] &= \\sum_{s} (X+Y)(s) P(s) \\\\\n", - " &= \\sum_{s} (X(s)P(s) + Y(s)P(s)) \\\\\n", - " &= \\sum_{s} X(s)P(s) + \\sum_{s} Y(s)P(s)\\\\\n", - " &= \\mathbb{E}[X] + \\mathbb{E}[Y]\n", - "\\end{align}$$\n", - ":::\n", - "\n", - "3. If $g$ is a non-linear function, then in general, \n", - "$$\\mathbb{E}[g(X)] \\neq g(\\mathbb{E}[X])$$\n", - "* For example, if $X$ is -1 or 1 with equal probability, then $\\mathbb{E}[X] = 0$ but $\\mathbb{E}[X^2] = 1 \\neq 0$\n", - "\n", - "### Properties of Variance\n", - "Recall the definition of variance: \n", - "$$\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2]$$\n", - "\n", - "1. Unlike expectation, variance is *non-linear*. The variance of the linear transformation $aX+b$ is:\n", - "$$\\text{Var}(aX+b) = a^2 \\text{Var}(X)$$\n", - "\n", - "* Subsequently, $$\\text{SD}(aX+b) = |a| \\text{SD}(X)$$\n", - "* The full proof of this fact can be found using the definition of variance. As general intuition, consider that $aX+b$ scales the variable $X$ by a factor of $a$, then shifts the distribution of $X$ by $b$ units. \n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Full Proof\n", - "We know that $$\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$$\n", - "\n", - "In order to compute $\\text{Var}(aX+b)$, consider that a shift by b units does not affect spread, so $\\text{Var}(aX+b) = \\text{Var}(aX)$\n", - "\n", - "Then, \n", - "$$\\begin{align}\n", - " \\text{Var}(aX+b) &= \\text{Var}(aX) \\\\\n", - " &= E((aX)^2) - (E(aX))^2\n", - " &= E(a^2 X^2) - (aE(X))^2\\\\\n", - " &= a^2 (E(X^2) - (E(X))^2) \\\\\n", - " &= a^2 \\text{Var}(X)\n", - "\\end{align}$$\n", - ":::\n", - "\n", - "* Shifting the distribution by $b$ *does not* impact the *spread* of the distribution. Thus, $\\text{Var}(aX+b) = \\text{Var}(aX)$.\n", - "* Scaling the distribution by $a$ *does* impact the spread of the distribution.\n", - "\n", - "

\n", - "transformation\n", - "

\n", - "\n", - "2. Variance of sums of RVs is affected by the (in)dependence of the RVs\n", - "$$\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) 2\\text{cov}(X,Y)$$\n", - "$$\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) \\qquad \\text{if } X, Y \\text{ independent}$$\n", - "\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "The variance of a sum is affected by the dependence between the two random variables that are being added. Let’s expand out the definition of $\\text{Var}(X + Y)$ to see what’s going on.\n", - "\n", - "To simplify the math, let $\\mu_x = \\mathbb{E}[X]$ and $\\mu_y = \\mathbb{E}[Y]$\n", - "\n", - "$$ \\begin{align}\n", - "\\text{Var}(X + Y) &= \\mathbb{E}[(X+Y- \\mathbb{E}(X+Y))^2] \\\\\n", - "&= \\mathbb{E}[((X - \\mu_x) + (Y - \\mu_y))^2] \\\\\n", - "&= \\mathbb{E}[(X - \\mu_x)^2 + 2(X - \\mu_x)(Y - \\mu_y) + (Y - \\mu_y)^2] \\\\\n", - "&= \\mathbb{E}[(X - \\mu_x)^2] + \\mathbb{E}[(Y - \\mu_y)^2] + \\mathbb{E}[(X - \\mu_x)(Y - \\mu_y)] \\\\\n", - "&= \\text{Var}(X) + \\text{Var}(Y) + \\mathbb{E}[(X - \\mu_x)(Y - \\mu_y)] \n", - "\\end{align}$$\n", - ":::\n", - "\n", - "### Covariance and Correlation\n", - "We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to *two* random variables: $\\text{Cov}(X, X) = \\mathbb{E}[(X - \\mathbb{E}[X])^2] = \\text{Var}(X)$.\n", - "\n", - "$$\\text{Cov}(X, Y) = \\mathbb{E}[(X - \\mathbb{E}[X])(Y - \\mathbb{E}[Y])]$$\n", - "\n", - "We can treat the covariance as a measure of association. Remember the definition of correlation given when we first established SLR?\n", - "\n", - "$$r(X, Y) = \\mathbb{E}\\left[\\left(\\frac{X-\\mathbb{E}[X]}{\\text{SD}(X)}\\right)\\left(\\frac{Y-\\mathbb{E}[Y]}{\\text{SD}(Y)}\\right)\\right] = \\frac{\\text{Cov}(X, Y)}{\\text{SD}(X)\\text{SD}(Y)}$$\n", - "\n", - "It turns out we've been quietly using covariance for some time now! If $X$ and $Y$ are independent, then $\\text{Cov}(X, Y) =0$ and $r(X, Y) = 0$. Note, however, that the converse is not always true: $X$ and $Y$ could have $\\text{Cov}(X, Y) = r(X, Y) = 0$ but not be independent. \n", - "\n", - "### Summary \n", - "* Let $X$ be a random variable with distribution $P(X=x). \n", - " * $\\mathbb{E}[X] = \\sum_{x} x P(X=x)$\n", - " * $\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2] = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2$\n", - "* Let $a$ and $b$ be scalar values. \n", - " * $\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$\n", - " * $\\text{Var}(aX+b) = a^2 \\text{Var}(X)$\n", - "* Let $Y$ be another random variable. \n", - " * $\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y]$\n", - " * $\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) 2\\text{cov}(X,Y)$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Common Random Variables\n", - "There are several cases of random variables that appear often and have useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable’s shape (i.e., distribution) and its values.\n", - "\n", - "* Bernoulli(p)\n", - " * Takes on value 1 with probability p, and 0 with probability 1 - p.\n", - " * AKA the “indicator” random variable.\n", - " * Let X be a Bernoulli(p) random variable\n", - " * $\\mathbb{E}[X] = 1 * p + 0 * (1-p) = p$\n", - " * $\\mathbb{E}[X^2] = 1^2 * p + 0 * (1-p) = p$\n", - " * $\\text{Var}(X) = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2 = p - p^2 = p(1-p)$\n", - "* Binomial(n, p)\n", - " * Number of 1s in 'n' independent Bernoulli(p) trials.\n", - " * Let $Y$ be a Binomial(n, p) random variable\n", - " * the distribution of $Y$ is given by the binomial formula, and we can write $Y = \\sum_{i=1}^n X_i$ where\n", - " * $X_i$ s the indicator of success on trial i. $X_i = 1$ if trial i is a success, else 0.\n", - " * all $X_i$ are i.i.d. and Bernoulli(p)\n", - " * $\\mathbb{E}[Y] = \\sum_{i=1}^n \\mathbb{E}[X_i] = np$\n", - " * $\\text{Var}(X) = \\sum_{i=1}^n \\text{Var}(X_i) = np(1-p)$ \n", - " * $X_i$'s are independent, so $\\text{Cov}(X_i, X_j) = 0$ for all i, j.\n", - "* Uniform on a finite set of values\n", - " * Probability of each value is 1 / (number of possible values).\n", - " * For example, a standard/fair die.\n", - "* Uniform on the unit interval (0, 1)\n", - " * Density is flat at 1 on (0, 1) and 0 elsewhere.\n", - "* Normal($\\mu, \\sigma^2$)\n", - " * $$f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left( -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^{\\!2}\\,\\right)$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Populations and Samples \n", - "Today, we've talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, variance, functions of the random variable, etc. \n", - "\n", - "In Data Science, however, we often do not have access to the whole population, so we don’t know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer properties of the population. \n", - "\n", - "When sampling, we make the (big) assumption that we sample uniformly at random with replacement from the population; each observation in our sample is a random variable drawn i.i.d from our population distribution. \n", - "\n", - "### Sample Mean \n", - "Consider an i.i.d. sample $X_1, X_2, ..., X_n$ drawn from a population with mean 𝜇 and SD 𝜎.\n", - "We define the sample mean as $$\\bar{X_n} = \\frac{1}{n} \\sum_{i=1}^n X_i$$\n", - "\n", - "The expectation of the sample mean is given by: \n", - "$$\\begin{align} \n", - " \\mathbb{E}[\\bar{X_n}] &= \\frac{1}{n} \\sum_{i=1}^n \\mathbb{E}[X_i] \\\\\n", - " &= \\frac{1}{n} (n \\mu) \\\\\n", - " &= \\mu \n", - "\\end{align}$$\n", - "\n", - "The variance is given by: \n", - "$$\\begin{align} \n", - " \\text{Var}(\\bar{X_n}) &= \\frac{1}{n^2} \\text{Var}( \\sum_{i=1}^n X_i) \\\\\n", - " &= \\frac{1}{n^2} \\left( \\sum_{i=1}^n \\text{Var}(X_i) \\right) \\\\\n", - " &= \\frac{1}{n^2} (n \\sigma^2) = \\frac{\\sigma^2}{n}\n", - "\\end{align}$$\n", - " \n", - "$\\bar{X_n}$ is normally distributed by the Central Limit Theorem (CLT).\n", - "\n", - "### Central Limit Theorem\n", - "The CLT states that no matter what population you are drawing from, if an i.i.d. sample of size $n$ is large, the probability distribution of the sample mean is roughly normal with mean 𝜇 and SD $\\sigma/\\sqrt{n}$.\n", - "\n", - "Any theorem that provides the rough distribution of a statistic and doesn’t need the distribution of the population is valuable to data scientists because we rarely know a lot about the population!\n", - "\n", - "For a more in-depth demo check out [onlinestatbook](https://onlinestatbook.com/stat_sim/sampling_dist/). \n", - "\n", - "THE CLT applies if the sample size $n$ is large, but how large does n have to be for the normal approximation to be good? It depends on the shape of the distribution of the population.\n", - "\n", - "* If population is roughly symmetric and unimodal/uniform, could need as few as $n = 20$.\n", - "* If population is very skewed, you will need bigger n.\n", - "* If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped.\n", - "\n", - "### Using the Sample Mean to Estimate the Population Mean\n", - "Our goal with sampling is often to estimate some characteristic of a population. When we collect a single sample, it has just one average. Since our sample was random, it *could* have come out differently. The CLT helps us understand this difference. We should consider the average value and spread of all possible sample means, and what this means for how big $n$ should be.\n", - "\n", - "For every sample size, the expected value of the sample mean is the population mean. $\\mathbb{E}[\\bar{X_n}] = \\mu$. We call the sample mean an unbiased estimator of the population mean, and we'll cover this more in next lecture. \n", - "\n", - "Square root law ([Data 8](https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html#the-square-root-law)) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. $\\text{SD}(\\bar{X_n}) = \\frac{\\sigma}{\\sqrt{n}}$. The sample mean is more likely to be close to the population mean if we have a larger sample size.\n", - "

\n", - "transformation\n", - "

" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - } + "cells": [ + { + "cell_type": "raw", + "metadata": {}, + "source": [ + "---\n", + "title: Random Variables\n", + "execute:\n", + " echo: true\n", + "format:\n", + " html:\n", + " code-fold: true\n", + " code-tools: true\n", + " toc: true\n", + " toc-title: Random Variables\n", + " page-layout: full\n", + " theme:\n", + " - cosmo\n", + " - cerulean\n", + " callout-icon: false\n", + "---" + ] }, - "nbformat": 4, - "nbformat_minor": 4 + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "::: {.callout-note collapse=\"false\"}\n", + "## Learning Outcomes\n", + "* Define a random variable in terms of its distribution\n", + "* Compute the expectation and variance of a random variable\n", + "* Gain familiarity with the Bernoulli and binomial random variables\n", + ":::\n", + "\n", + "In the past few lectures, we've examined the role of complexity in influencing model performance. We've considered model complexity in the context of a tradeoff between two competing factors: model variance and training error. \n", + "\n", + "Thus far, our analysis has been mostly qualitative. We've acknowledged that our choice of model complexity needs to strike a balance between model variance and training error, but we haven't yet discussed *why* exactly this tradeoff exists.\n", + "\n", + "To better understand the origin of this tradeoff, we will need to introduce the language of **random variables**. The next two lectures on probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called **bias-variance tradeoff**. Our roadmap for the next few lectures will be:\n", + "\n", + "1. Random Variables Estimators: introduce random variables, considering the concepts of expectation, variance, and covariance\n", + "2. Estimators, Bias, and Variance: re-express the ideas of model variance and training error in terms of random variables and use this new perspective to investigate our choice of model complexity\n", + "\n", + "::: {.callout-tip}\n", + "## Data 8\n", + "Recall the following concepts from Data 8: \n", + "\n", + "1. Sample mean: the mean of your random sample\n", + "2. Central Limit Theorem: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean\n", + "\n", + " a. is roughly normal\n", + "\n", + " b. is centered at the population mean\n", + " \n", + " c. has an $SD = \\frac{\\text{population SD}}{\\sqrt{\\text{sample size}}}$\n", + "\n", + ":::" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Random Variables and Distributions\n", + "\n", + "Suppose we generate a set of random data, like a random sample from some population. A **random variable** is a *numerical function* of the randomness in the data. It is *random* since our sample was drawn at random; it is *variable* because its exact value depends on how this random sample came out. As such, the domain or input of our random variable is all possible (random) outcomes in a *sample space*, and its range or output is the number line. We typically denote random variables with uppercase letters, such as $X$ or $Y$. \n", + "\n", + "### Distribution\n", + "For any random variable $X$, we need to be able to specify 2 things: \n", + "1. Possible values: the set of values the random variable can take on.\n", + "2. Probabilities: the set of probabilities describing how the total probability of 100% is split over the possible values.\n", + "\n", + "If $X$ is discrete (has a finite number of possible values),\n", + "* The probability that a random variable $X$ takes on the value $x$ is given by $P(X=x)$.\n", + "* Probabilities must sum to 1: $\\sum_{\\text{all} x} P(X=x) = 1$,\n", + "\n", + "We can often display this using a **probability distribution table** (example shown below).\n", + "\n", + "The **distribution** of a random variable $X$ is a description of how the total probability of 100% is split over all the possible values of $X$, and it fully defines a random variable.\n", + "\n", + "The distribution of a discrete random variable can also be represented using a histogram. If a variable is **continuous** – it can take on infinitely many values – we can illustrate its distribution using a density curve. \n", + "\n", + "

\n", + "discrete_continuous\n", + "

\n", + "\n", + "Probabilities are areas. For discrete random variables, the *area of the red bars* represent the probability that a discrete random variable $X$ falls within those values. For continuous random variables, the *area under the curve* represents the probability that a discrete random variable $Y$ falls within those values.\n", + "\n", + "

\n", + "discrete_continuous\n", + "

\n", + "\n", + "If we sum up the total area of the bars/under the density curve, we should get 100%, or 1.\n", + "\n", + "\n", + "### Example: Tossing a Coin\n", + "To give a concrete example, let's formally define a fair coin toss. A fair coin can land on heads ($H$) or tails ($T$), each with a probability of 0.5. With these possible outcomes, we can define a random variable $X$ as: \n", + "$$X = \\begin{cases} \n", + " 1, \\text{if the coin lands heads} \\\\\n", + " 0, \\text{if the coin lands tails} \n", + " \\end{cases}$$\n", + "\n", + "$X$ is a function with a domain (input) of $\\{H, T\\}$ and a range (output) of $\\{1, 0\\}$. We can write this in function notation as \n", + "$$\\begin{cases} X(H) = 1 \\\\ X(T) = 0 \\end{cases}$$\n", + "and the probability distribution table of $X$ is \n", + "\n", + "| $x$ | $P(X=x)$ | \n", + "| --- | -------- |\n", + "| 0 | $\\frac{1}{2}$ | \n", + "| 1 | $\\frac{1}{2}$ |\n", + "\n", + "Suppose we draw a random sample $s$ of size 3 from all students enrolled in Data 100. \n", + "We can define $Y$ as the number of data science students in our sample. Its domain is all possible samples of size 3, and its range is $\\{0, 1, 2, 3\\}$.\n", + "\n", + "

\n", + " rv\n", + "

\n", + "\n", + "We can show the distribution of $Y$ in the following tables. The table on the left lists all possible samples of $s$ and the number of times they can appear ($Y(s)$). We can use this to calculate the values for the table on the right, a **probability distribution table**. \n", + "\n", + "

\n", + "distribution\n", + "

\n", + "\n", + "### Simulation\n", + "Given a random variable $X$’s distribution, how could we **generate/simulate** a population? To do so, we can randomly pick values of $X$ according to its distribution using `np.random.choice` or `df.sample`. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Expectation and Variance\n", + "There are several ways to describe a random variable. The methods shown above - table of all samples $s, X(s)$, distribution table $P(X=x)$, and histograms - are all definitions that fully describe a random variable. Often, it is easier to describe a random variable using some numerical summary, rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a \"summary\" of how the variable tends to behave, they are *not* random – think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable.\n", + "\n", + "### Expectation\n", + "The **expectation** of a random variable $X$ is the weighted average of the values of $X$, where the weights are the probabilities of each value occurring. There are two equivalent ways to compute the expectation: \n", + "1. Apply the weights one *sample* at a time: $$\\mathbb{E}[X] = \\sum_{\\text{all possible } s} X(s) P(s)$$.\n", + "2. Apply the weights one possible *value* at a time: $$\\mathbb{E}[X] = \\sum_{\\text{all possible } x} x P(X=x)$$\n", + "\n", + "We want to emphasize that the expectation is a *number*, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram. If we simulate the variable many times, it is the long-run average of the random variable.\n", + "\n", + "### Example 1: Coin Toss\n", + "Going back to our coin toss example, we define a random variable $X$ as: \n", + "$$X = \\begin{cases} \n", + " 1, \\text{if the coin lands heads} \\\\\n", + " 0, \\text{if the coin lands tails} \n", + " \\end{cases}$$\n", + "We can calculate its expectation $\\mathbb{E}[X]$ using the second method of applying the weights one possible value at a time: \n", + "$$\\begin{align}\n", + " \\mathbb{E}[X] &= \\sum_{x} x P(X=x) \\\\\n", + " &= 1 * 0.5 + 0 * 0.5 \\\\\n", + " &= 0.5\n", + "\\end{align}$$\n", + "Note that $\\mathbb{E}[X] = 0.5$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**.\n", + "\n", + "### Example 2\n", + "Consider the random variable $X$: \n", + "\n", + "| $x$ | $P(X=x)$ | \n", + "| --- | -------- |\n", + "| 3 | 0.1 | \n", + "| 4 | 0.2 |\n", + "| 6 | 0.4 | \n", + "| 8 | 0.3 |\n", + "\n", + "To calculate it's expectation, \n", + "$$\\begin{align}\n", + " \\mathbb{E}[X] &= \\sum_{x} x P(X=x) \\\\\n", + " &= 3 * 0.1 + 4 * 0.2 + 6 * 0.4 + 8 * 0.3 \\\\\n", + " &= 0.3 + 0.8 + 2.4 + 2.4 \\\\\n", + " &= 5.9\n", + "\\end{align}$$\n", + "Again, note that $\\mathbb{E}[X] = 5.9$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**.\n", + "\n", + "### Variance\n", + "The **variance** of a random variable is a measure of its chance error. It is defined as the expected squared deviation from the expectation of $X$. Put more simply, variance asks: how far does $X$ typically vary from its average value, just by chance? What is the spread of $X$'s distribution?\n", + "\n", + "$$\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2]$$\n", + "\n", + "The units of variance are the square of the units of $X$. To get it back to the right scale, use the standard deviation of $X$: $\\text{SD}(X) = \\sqrt{\\text{Var}(X)}$.\n", + "\n", + "Like with expectation, **variance is a number, not a random variable**! Its main use is to quantify chance error.\n", + "\n", + "By [Chebyshev’s inequality](https://www.inferentialthinking.com/chapters/14/2/Variability.html#Chebychev's-Bounds), which you saw in Data 8, no matter what the shape of the distribution of X is, the vast majority of the probability lies in the interval “expectation plus or minus a few SDs.”\n", + "\n", + "If we expand the square and use properties of expectation, we can re-express variance as the **computational formula for variance**. This form is often more convenient to use when computing the variance of a variable by hand, and it is also useful in Mean Squared Error calculations (if $X$ is centered and $E(X)=0$, then $\\mathbb{E}[X^2] = \\text{Var}(X)$).\n", + "\n", + "$$\\text{Var}(X) = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2$$\n", + "\n", + "
\n", + " Proof (toggle this cell)\n", + " \n", + "$$\\begin{align}\n", + " \\text{Var}(X) &= \\mathbb{E}[(X-\\mathbb{E}[X])^2] \\\\\n", + " &= \\mathbb{E}(X^2 - 2X\\mathbb{E}(X) + (\\mathbb{E}(X))^2) \\\\\n", + " &= \\mathbb{E}(X^2) - 2 \\mathbb{E}(X)\\mathbb{E}(X) +( \\mathbb{E}(X))^2\\\\\n", + " &= \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2\n", + " \\end{align}$$\n", + " \n", + "
\n", + "\n", + "How do we compute $\\mathbb{E}[X^2]$? Any function of a random variable is *also* a random variable – that means that by squaring $X$, we've created a new random variable. To compute $\\mathbb{E}[X^2]$, we can simply apply our definition of expectation to the random variable $X^2$.\n", + "\n", + "$$\\mathbb{E}[X^2] = \\sum_{x} x^2 P(X = x)$$ \n", + "\n", + "### Example: Dice\n", + "Let $X$ be the outcome of a single fair dice roll. $X$ is a random variable defined as \n", + "$$X = \\begin{cases} \n", + " \\frac{1}{6}, \\text{if } x \\in \\{1,2,3,4,5,6\\} \\\\\n", + " 0, \\text{otherwise} \n", + " \\end{cases}$$\n", + "\n", + "::: {.callout-caution collapse=\"true\"}\n", + "## What's the expectation $\\mathbb{E}[X]?$\n", + "\n", + "$$ \\begin{align} \n", + " \\mathbb{E}[X] &= 1(\\frac{1}{6}) + 2(\\frac{1}{6}) + 3(\\frac{1}{6}) + 4(\\frac{1}{6}) + 5(\\frac{1}{6}) + 6(\\frac{1}{6}) \\\\\n", + " &= (\\frac{1}{6}) ( 1 + 2 + 3 + 4 + 5 + 6) \\\\\n", + " &= \\frac{7}{2}\n", + " \\end{align}$$\n", + ":::\n", + "\n", + "::: {.callout-caution collapse=\"true\"}\n", + "## What's the variance $\\text{Var}(X)?$\n", + "\n", + "Using approach 1: \n", + " $$\\begin{align} \n", + " \\text{Var}(X) &= (\\frac{1}{6})((1 - \\frac{7}{2})^2 + (2 - \\frac{7}{2})^2 + (3 - \\frac{7}{2})^2 + (4 - \\frac{7}{2})^2 + (5 - \\frac{7}{2})^2 + (6 - \\frac{7}{2})^2) \\\\\n", + " &= \\frac{35}{12}\n", + " \\end{align}$$\n", + "\n", + "Using approach 2: \n", + "$$\\mathbb{E}[X^2] = \\sum_{x} x^2 P(X = x) = \\frac{91}{6}$$\n", + "$$\\text{Var}(X) = \\frac{91}{6} - (\\frac{7}{2})^2 = \\frac{35}{12}$$\n", + ":::" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Sums of Random Variables\n", + "Often, we will work with multiple random variables at the same time. A function of a random variable is also a random variable, so if you create multiple random variables based on your sample, then functions of those random variables are also random variables.\n", + "\n", + "For example, if $X_1, X_2, ..., X_n$ are random variables, then so are all of these: \n", + "* $X_n^2$\n", + "* $\\#\\{i : X_i > 10\\}$\n", + "* $\\text{max}(X_1, X_2, ..., X_n)$\n", + "* $\\frac{1}{n} \\sum_{i=1}^n (X_i - c)^2$\n", + "* $\\frac{1}{n} \\sum_{i=1}^n X_i$\n", + "\n", + "\n", + "### Equal vs. Identically Distributed vs. i.i.d\n", + "Suppose that we have two random variables $X$ and $Y$:\n", + "\n", + "* $X$ and $Y$ are **equal** if $X(s) = Y(s)$ for every sample $s$. Regardless of the exact sample drawn, $X$ is always equal to $Y$.\n", + "* $X$ and $Y$ are **identically distributed** if the distribution of $X$ is equal to the distribution of $Y$. We say “X and Y are equal in distribution.” That is, $X$ and $Y$ take on the same set of possible values, and each of these possible values is taken with the same probability. On any specific sample $s$, identically distributed variables do *not* necessarily share the same value. If X = Y, then X and Y are identically distributed; however, the converse is not true (ex: Y = 7-X, X is a die)\n", + "* $X$ and $Y$ are **independent and identically distributed (i.i.d)** if \n", + " 1. The variables are identically distributed. \n", + " 2. Knowing the outcome of one variable does not influence our belief of the outcome of the other.\n", + "\n", + "For example, let $X_1$ and $X_2$ be numbers on rolls of two fair die. $X_1$ and $X_2$ are i.i.d, so $X_1$ and $X_2$ have the same distribution. However, the sums $Y = X_1 + X_1 = 2X_1$ and $Z=X_1+X_2$ have different distributions but the same expectation.\n", + "\n", + "

\n", + "distribution\n", + "

\n", + "\n", + "However, $Y = X_1$ has a larger variance\n", + "\n", + "

\n", + "distribution\n", + "

\n", + "\n", + "### Properties of Expectation \n", + "Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: $$\\mathbb{E}[X] = \\sum_{x} x P(X=x)$$\n", + "\n", + "1. **Linearity of expectation**. The expectation of the linear transformation $aX+b$, where $a$ and $b$ are constants, is:\n", + "\n", + "$$\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$$\n", + "\n", + "::: {.callout-tip collapse=\"true\"}\n", + "## Proof\n", + "$$\\begin{align}\n", + " \\mathbb{E}[aX+b] &= \\sum_{x} (ax + b) P(X=x) \\\\\n", + " &= \\sum_{x} (ax P(X=x) + bP(X=x)) \\\\\n", + " &= a\\sum_{x}P(X=x) + b\\sum_{x}P(X=x)\\\\\n", + " &= a\\mathbb{E}(X) = b * 1\n", + " \\end{align}$$\n", + ":::\n", + "\n", + "2. Expectation is also linear in *sums* of random variables. \n", + "\n", + "$$\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y]$$\n", + "\n", + "::: {.callout-tip collapse=\"true\"}\n", + "## Proof\n", + "$$\\begin{align}\n", + " \\mathbb{E}[X+Y] &= \\sum_{s} (X+Y)(s) P(s) \\\\\n", + " &= \\sum_{s} (X(s)P(s) + Y(s)P(s)) \\\\\n", + " &= \\sum_{s} X(s)P(s) + \\sum_{s} Y(s)P(s)\\\\\n", + " &= \\mathbb{E}[X] + \\mathbb{E}[Y]\n", + "\\end{align}$$\n", + ":::\n", + "\n", + "3. If $g$ is a non-linear function, then in general, \n", + "$$\\mathbb{E}[g(X)] \\neq g(\\mathbb{E}[X])$$\n", + "\n", + "* For example, if $X$ is -1 or 1 with equal probability, then $\\mathbb{E}[X] = 0$ but $\\mathbb{E}[X^2] = 1 \\neq 0$\n", + "\n", + "### Properties of Variance\n", + "Recall the definition of variance: \n", + "$$\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2]$$\n", + "\n", + "1. Unlike expectation, variance is *non-linear*. The variance of the linear transformation $aX+b$ is:\n", + "$$\\text{Var}(aX+b) = a^2 \\text{Var}(X)$$\n", + "\n", + "* Subsequently, $$\\text{SD}(aX+b) = |a| \\text{SD}(X)$$\n", + "* The full proof of this fact can be found using the definition of variance. As general intuition, consider that $aX+b$ scales the variable $X$ by a factor of $a$, then shifts the distribution of $X$ by $b$ units. \n", + "\n", + "::: {.callout-tip collapse=\"true\"}\n", + "## Full Proof\n", + "We know that $$\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$$\n", + "\n", + "In order to compute $\\text{Var}(aX+b)$, consider that a shift by b units does not affect spread, so $\\text{Var}(aX+b) = \\text{Var}(aX)$.\n", + "\n", + "Then, \n", + "$$\\begin{align}\n", + " \\text{Var}(aX+b) &= \\text{Var}(aX) \\\\\n", + " &= E((aX)^2) - (E(aX))^2\n", + " &= E(a^2 X^2) - (aE(X))^2\\\\\n", + " &= a^2 (E(X^2) - (E(X))^2) \\\\\n", + " &= a^2 \\text{Var}(X)\n", + "\\end{align}$$\n", + ":::\n", + "\n", + "* Shifting the distribution by $b$ *does not* impact the *spread* of the distribution. Thus, $\\text{Var}(aX+b) = \\text{Var}(aX)$.\n", + "* Scaling the distribution by $a$ *does* impact the spread of the distribution.\n", + "\n", + "

\n", + "transformation\n", + "

\n", + "\n", + "2. Variance of sums of RVs is affected by the (in)dependence of the RVs.\n", + "$$\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) 2\\text{cov}(X,Y)$$\n", + "$$\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) \\qquad \\text{if } X, Y \\text{ independent}$$\n", + "\n", + "\n", + "::: {.callout-tip collapse=\"true\"}\n", + "## Proof\n", + "The variance of a sum is affected by the dependence between the two random variables that are being added. Let’s expand out the definition of $\\text{Var}(X + Y)$ to see what’s going on.\n", + "\n", + "To simplify the math, let $\\mu_x = \\mathbb{E}[X]$ and $\\mu_y = \\mathbb{E}[Y]$.\n", + "\n", + "$$ \\begin{align}\n", + "\\text{Var}(X + Y) &= \\mathbb{E}[(X+Y- \\mathbb{E}(X+Y))^2] \\\\\n", + "&= \\mathbb{E}[((X - \\mu_x) + (Y - \\mu_y))^2] \\\\\n", + "&= \\mathbb{E}[(X - \\mu_x)^2 + 2(X - \\mu_x)(Y - \\mu_y) + (Y - \\mu_y)^2] \\\\\n", + "&= \\mathbb{E}[(X - \\mu_x)^2] + \\mathbb{E}[(Y - \\mu_y)^2] + \\mathbb{E}[(X - \\mu_x)(Y - \\mu_y)] \\\\\n", + "&= \\text{Var}(X) + \\text{Var}(Y) + \\mathbb{E}[(X - \\mu_x)(Y - \\mu_y)] \n", + "\\end{align}$$\n", + ":::\n", + "\n", + "### Covariance and Correlation\n", + "We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to *two* random variables: $\\text{Cov}(X, X) = \\mathbb{E}[(X - \\mathbb{E}[X])^2] = \\text{Var}(X)$\n", + "\n", + "$$\\text{Cov}(X, Y) = \\mathbb{E}[(X - \\mathbb{E}[X])(Y - \\mathbb{E}[Y])]$$\n", + "\n", + "We can treat the covariance as a measure of association. Remember the definition of correlation given when we first established SLR?\n", + "\n", + "$$r(X, Y) = \\mathbb{E}\\left[\\left(\\frac{X-\\mathbb{E}[X]}{\\text{SD}(X)}\\right)\\left(\\frac{Y-\\mathbb{E}[Y]}{\\text{SD}(Y)}\\right)\\right] = \\frac{\\text{Cov}(X, Y)}{\\text{SD}(X)\\text{SD}(Y)}$$\n", + "\n", + "It turns out we've been quietly using covariance for some time now! If $X$ and $Y$ are independent, then $\\text{Cov}(X, Y) =0$ and $r(X, Y) = 0$. Note, however, that the converse is not always true: $X$ and $Y$ could have $\\text{Cov}(X, Y) = r(X, Y) = 0$ but not be independent. \n", + "\n", + "### Summary \n", + "* Let $X$ be a random variable with distribution $P(X=x)$. \n", + " * $\\mathbb{E}[X] = \\sum_{x} x P(X=x)$\n", + " * $\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2] = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2$\n", + "* Let $a$ and $b$ be scalar values. \n", + " * $\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$\n", + " * $\\text{Var}(aX+b) = a^2 \\text{Var}(X)$\n", + "* Let $Y$ be another random variable. \n", + " * $\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y]$\n", + " * $\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) 2\\text{cov}(X,Y)$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Common Random Variables\n", + "There are several cases of random variables that appear often and have useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable’s shape (i.e., distribution) and its values.\n", + "\n", + "* Bernoulli(p)\n", + " * Takes on value 1 with probability p, and 0 with probability 1 - p.\n", + " * AKA the “indicator” random variable.\n", + " * Let X be a Bernoulli(p) random variable\n", + " * $\\mathbb{E}[X] = 1 * p + 0 * (1-p) = p$\n", + " * $\\mathbb{E}[X^2] = 1^2 * p + 0 * (1-p) = p$\n", + " * $\\text{Var}(X) = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2 = p - p^2 = p(1-p)$\n", + "* Binomial(n, p)\n", + " * Number of 1s in 'n' independent Bernoulli(p) trials.\n", + " * Let $Y$ be a Binomial(n, p) random variable.\n", + " * The distribution of $Y$ is given by the binomial formula, and we can write $Y = \\sum_{i=1}^n X_i$ where:\n", + " * $X_i$ s the indicator of success on trial i. $X_i = 1$ if trial i is a success, else 0.\n", + " * All $X_i$ are i.i.d. and Bernoulli(p).\n", + " * $\\mathbb{E}[Y] = \\sum_{i=1}^n \\mathbb{E}[X_i] = np$\n", + " * $\\text{Var}(X) = \\sum_{i=1}^n \\text{Var}(X_i) = np(1-p)$ \n", + " * $X_i$'s are independent, so $\\text{Cov}(X_i, X_j) = 0$ for all i, j.\n", + "* Uniform on a finite set of values\n", + " * Probability of each value is 1 / (number of possible values).\n", + " * For example, a standard/fair die.\n", + "* Uniform on the unit interval (0, 1)\n", + " * Density is flat at 1 on (0, 1) and 0 elsewhere.\n", + "* Normal($\\mu, \\sigma^2$)\n", + " * $$f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left( -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^{\\!2}\\,\\right)$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Populations and Samples \n", + "Today, we've talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, variance, functions of the random variable, etc. \n", + "\n", + "In Data Science, however, we often do not have access to the whole population, so we don’t know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer properties of the population. \n", + "\n", + "When sampling, we make the (big) assumption that we sample uniformly at random with replacement from the population; each observation in our sample is a random variable drawn i.i.d from our population distribution. \n", + "\n", + "### Sample Mean \n", + "Consider an i.i.d. sample $X_1, X_2, ..., X_n$ drawn from a population with mean 𝜇 and SD 𝜎.\n", + "We define the sample mean as $$\\bar{X_n} = \\frac{1}{n} \\sum_{i=1}^n X_i$$\n", + "\n", + "The expectation of the sample mean is given by: \n", + "$$\\begin{align} \n", + " \\mathbb{E}[\\bar{X_n}] &= \\frac{1}{n} \\sum_{i=1}^n \\mathbb{E}[X_i] \\\\\n", + " &= \\frac{1}{n} (n \\mu) \\\\\n", + " &= \\mu \n", + "\\end{align}$$\n", + "\n", + "The variance is given by: \n", + "$$\\begin{align} \n", + " \\text{Var}(\\bar{X_n}) &= \\frac{1}{n^2} \\text{Var}( \\sum_{i=1}^n X_i) \\\\\n", + " &= \\frac{1}{n^2} \\left( \\sum_{i=1}^n \\text{Var}(X_i) \\right) \\\\\n", + " &= \\frac{1}{n^2} (n \\sigma^2) = \\frac{\\sigma^2}{n}\n", + "\\end{align}$$\n", + " \n", + "$\\bar{X_n}$ is normally distributed by the Central Limit Theorem (CLT).\n", + "\n", + "### Central Limit Theorem\n", + "The CLT states that no matter what population you are drawing from, if an i.i.d. sample of size $n$ is large, the probability distribution of the sample mean is roughly normal with mean 𝜇 and SD $\\sigma/\\sqrt{n}$.\n", + "\n", + "Any theorem that provides the rough distribution of a statistic and doesn’t need the distribution of the population is valuable to data scientists because we rarely know a lot about the population!\n", + "\n", + "For a more in-depth demo check out [onlinestatbook](https://onlinestatbook.com/stat_sim/sampling_dist/). \n", + "\n", + "The CLT applies if the sample size $n$ is large, but how large does n have to be for the normal approximation to be good? It depends on the shape of the distribution of the population.\n", + "\n", + "* If the population is roughly symmetric and unimodal/uniform, could need as few as $n = 20$.\n", + "* If the population is very skewed, you will need a bigger n.\n", + "* If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped.\n", + "\n", + "### Using the Sample Mean to Estimate the Population Mean\n", + "Our goal with sampling is often to estimate some characteristic of a population. When we collect a single sample, it has just one average. Since our sample was random, it *could* have come out differently. The CLT helps us understand this difference. We should consider the average value and spread of all possible sample means, and what this means for how big $n$ should be.\n", + "\n", + "For every sample size, the expected value of the sample mean is the population mean. $\\mathbb{E}[\\bar{X_n}] = \\mu$. We call the sample mean an unbiased estimator of the population mean, and we'll cover this more in the next lecture. \n", + "\n", + "The square root law ([Data 8](https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html#the-square-root-law)) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. $\\text{SD}(\\bar{X_n}) = \\frac{\\sigma}{\\sqrt{n}}$. The sample mean is more likely to be close to the population mean if we have a larger sample size.\n", + "\n", + "

\n", + "transformation\n", + "

" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.16" + } + }, + "nbformat": 4, + "nbformat_minor": 4 } diff --git a/probability_1/probability_1.qmd b/probability_1/probability_1.qmd index 92a45023..652a2544 100644 --- a/probability_1/probability_1.qmd +++ b/probability_1/probability_1.qmd @@ -16,7 +16,7 @@ format: jupyter: python3 --- -::: {.callout-note collapse="true"} +::: {.callout-note collapse="false"} ## Learning Outcomes * Define a random variable in terms of its distribution * Compute the expectation and variance of a random variable @@ -27,7 +27,7 @@ In the past few lectures, we've examined the role of complexity in influencing m Thus far, our analysis has been mostly qualitative. We've acknowledged that our choice of model complexity needs to strike a balance between model variance and training error, but we haven't yet discussed *why* exactly this tradeoff exists. -To better understand the origin of this tradeoff, we will need to introduce the language of **random variables**. The next two lectures of probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called **bias-variance tradeoff**. Our roadmap for the next few lectures will be: +To better understand the origin of this tradeoff, we will need to introduce the language of **random variables**. The next two lectures on probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called **bias-variance tradeoff**. Our roadmap for the next few lectures will be: 1. Random Variables Estimators: introduce random variables, considering the concepts of expectation, variance, and covariance 2. Estimators, Bias, and Variance: re-express the ideas of model variance and training error in terms of random variables and use this new perspective to investigate our choice of model complexity @@ -36,7 +36,7 @@ To better understand the origin of this tradeoff, we will need to introduce the ## Data 8 Recall the following concepts from Data 8: -1. sample mean: the mean of your random sample +1. Sample mean: the mean of your random sample 2. Central Limit Theorem: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean a. is roughly normal @@ -53,29 +53,34 @@ Suppose we generate a set of random data, like a random sample from some populat ### Distribution For any random variable $X$, we need to be able to specify 2 things: -1. possible values: the set of values the random variable can take on -2. probabilities: the set of probabilities describing how the total probability of 100% is split over the possible values. +1. Possible values: the set of values the random variable can take on. +2. Probabilities: the set of probabilities describing how the total probability of 100% is split over the possible values. If $X$ is discrete (has a finite number of possible values), * The probability that a random variable $X$ takes on the value $x$ is given by $P(X=x)$. -* Probabilities must sum to 1: $\sum_{\text{all} x} P(X=x) = 1$ -* we can often do this using a **probability distribution table** (example shown below). +* Probabilities must sum to 1: $\sum_{\text{all} x} P(X=x) = 1$, + +We can often display this using a **probability distribution table** (example shown below). The **distribution** of a random variable $X$ is a description of how the total probability of 100% is split over all the possible values of $X$, and it fully defines a random variable. The distribution of a discrete random variable can also be represented using a histogram. If a variable is **continuous** – it can take on infinitely many values – we can illustrate its distribution using a density curve. +

discrete_continuous

+ Probabilities are areas. For discrete random variables, the *area of the red bars* represent the probability that a discrete random variable $X$ falls within those values. For continuous random variables, the *area under the curve* represents the probability that a discrete random variable $Y$ falls within those values. +

discrete_continuous

+ If we sum up the total area of the bars/under the density curve, we should get 100%, or 1. ### Example: Tossing a Coin -To give a concrete example, let's formally define a fair coin toss. A fair coin can land on heads ($H$) or Tails ($T$), each with a probability of 0.5. With these possible outcomes, we can define a random variable $X$ as: +To give a concrete example, let's formally define a fair coin toss. A fair coin can land on heads ($H$) or tails ($T$), each with a probability of 0.5. With these possible outcomes, we can define a random variable $X$ as: $$X = \begin{cases} 1, \text{if the coin lands heads} \\ 0, \text{if the coin lands tails} @@ -91,13 +96,14 @@ and the probability distribution table of $X$ is | 1 | $\frac{1}{2}$ | Suppose we draw a random sample $s$ of size 3 from all students enrolled in Data 100. -We can define $Y$ as the number of data science students in our sample. It's domain is all possible samples of size 3, and its range is $\{0, 1, 2, 3\}$. +We can define $Y$ as the number of data science students in our sample. Its domain is all possible samples of size 3, and its range is $\{0, 1, 2, 3\}$.

rv

We can show the distribution of $Y$ in the following tables. The table on the left lists all possible samples of $s$ and the number of times they can appear ($Y(s)$). We can use this to calculate the values for the table on the right, a **probability distribution table**. +

distribution

@@ -106,14 +112,14 @@ We can show the distribution of $Y$ in the following tables. The table on the le Given a random variable $X$’s distribution, how could we **generate/simulate** a population? To do so, we can randomly pick values of $X$ according to its distribution using `np.random.choice` or `df.sample`. ## Expectation and Variance -There are several ways to describe a random variable. The methods shown above - table of all samples $s, X(s)$, distribution table $P(X=x)$, histograms - are all definitions that fully describe a random variable. Often, it is easier to describe a random variable using some numerical summary, rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a "summary" of how the variable tends to behave, they are *not* random – think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable. +There are several ways to describe a random variable. The methods shown above - table of all samples $s, X(s)$, distribution table $P(X=x)$, and histograms - are all definitions that fully describe a random variable. Often, it is easier to describe a random variable using some numerical summary, rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a "summary" of how the variable tends to behave, they are *not* random – think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable. ### Expectation The **expectation** of a random variable $X$ is the weighted average of the values of $X$, where the weights are the probabilities of each value occurring. There are two equivalent ways to compute the expectation: 1. Apply the weights one *sample* at a time: $$\mathbb{E}[X] = \sum_{\text{all possible } s} X(s) P(s)$$. 2. Apply the weights one possible *value* at a time: $$\mathbb{E}[X] = \sum_{\text{all possible } x} x P(X=x)$$ -We want to emphasize that the expectation is a *number*, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram. If we simulate the variable many times, it is the long run average of the random variable. +We want to emphasize that the expectation is a *number*, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram. If we simulate the variable many times, it is the long-run average of the random variable. ### Example 1: Coin Toss Going back to our coin toss example, we define a random variable $X$ as: @@ -121,7 +127,7 @@ $$X = \begin{cases} 1, \text{if the coin lands heads} \\ 0, \text{if the coin lands tails} \end{cases}$$ -We can calculate the its expectation $\mathbb{E}[X]$ using the second method of applying the weights one possible value at a time: +We can calculate its expectation $\mathbb{E}[X]$ using the second method of applying the weights one possible value at a time: $$\begin{align} \mathbb{E}[X] &= \sum_{x} x P(X=x) \\ &= 1 * 0.5 + 0 * 0.5 \\ @@ -155,7 +161,7 @@ $$\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2]$$ The units of variance are the square of the units of $X$. To get it back to the right scale, use the standard deviation of $X$: $\text{SD}(X) = \sqrt{\text{Var}(X)}$. -Like with expectation, **variance is a number, not a random variable**! It's main use is to quantify chance error. +Like with expectation, **variance is a number, not a random variable**! Its main use is to quantify chance error. By [Chebyshev’s inequality](https://www.inferentialthinking.com/chapters/14/2/Variability.html#Chebychev's-Bounds), which you saw in Data 8, no matter what the shape of the distribution of X is, the vast majority of the probability lies in the interval “expectation plus or minus a few SDs.” @@ -166,12 +172,13 @@ $$\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$$
Proof (toggle this cell) - $$\begin{align} +$$\begin{align} \text{Var}(X) &= \mathbb{E}[(X-\mathbb{E}[X])^2] \\ &= \mathbb{E}(X^2 - 2X\mathbb{E}(X) + (\mathbb{E}(X))^2) \\ &= \mathbb{E}(X^2) - 2 \mathbb{E}(X)\mathbb{E}(X) +( \mathbb{E}(X))^2\\ &= \mathbb{E}[X^2] - (\mathbb{E}[X])^2 \end{align}$$ +
How do we compute $\mathbb{E}[X^2]$? Any function of a random variable is *also* a random variable – that means that by squaring $X$, we've created a new random variable. To compute $\mathbb{E}[X^2]$, we can simply apply our definition of expectation to the random variable $X^2$. @@ -226,14 +233,17 @@ Suppose that we have two random variables $X$ and $Y$: * $X$ and $Y$ are **equal** if $X(s) = Y(s)$ for every sample $s$. Regardless of the exact sample drawn, $X$ is always equal to $Y$. * $X$ and $Y$ are **identically distributed** if the distribution of $X$ is equal to the distribution of $Y$. We say “X and Y are equal in distribution.” That is, $X$ and $Y$ take on the same set of possible values, and each of these possible values is taken with the same probability. On any specific sample $s$, identically distributed variables do *not* necessarily share the same value. If X = Y, then X and Y are identically distributed; however, the converse is not true (ex: Y = 7-X, X is a die) * $X$ and $Y$ are **independent and identically distributed (i.i.d)** if - 1. the variables are identically distributed and - 2. knowing the outcome of one variable does not influence our belief of the outcome of the other. + 1. The variables are identically distributed. + 2. Knowing the outcome of one variable does not influence our belief of the outcome of the other. For example, let $X_1$ and $X_2$ be numbers on rolls of two fair die. $X_1$ and $X_2$ are i.i.d, so $X_1$ and $X_2$ have the same distribution. However, the sums $Y = X_1 + X_1 = 2X_1$ and $Z=X_1+X_2$ have different distributions but the same expectation. +

distribution

+ However, $Y = X_1$ has a larger variance +

distribution

@@ -241,7 +251,7 @@ However, $Y = X_1$ has a larger variance ### Properties of Expectation Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: $$\mathbb{E}[X] = \sum_{x} x P(X=x)$$ -1.**Linearity of expectation**. The expectation of the linear transformation $aX+b$, where $a$ and $b$ are constants, is: +1. **Linearity of expectation**. The expectation of the linear transformation $aX+b$, where $a$ and $b$ are constants, is: $$\mathbb{E}[aX+b] = aE[\mathbb{X}] + b$$ @@ -271,6 +281,7 @@ $$\begin{align} 3. If $g$ is a non-linear function, then in general, $$\mathbb{E}[g(X)] \neq g(\mathbb{E}[X])$$ + * For example, if $X$ is -1 or 1 with equal probability, then $\mathbb{E}[X] = 0$ but $\mathbb{E}[X^2] = 1 \neq 0$ ### Properties of Variance @@ -287,7 +298,7 @@ $$\text{Var}(aX+b) = a^2 \text{Var}(X)$$ ## Full Proof We know that $$\mathbb{E}[aX+b] = aE[\mathbb{X}] + b$$ -In order to compute $\text{Var}(aX+b)$, consider that a shift by b units does not affect spread, so $\text{Var}(aX+b) = \text{Var}(aX)$ +In order to compute $\text{Var}(aX+b)$, consider that a shift by b units does not affect spread, so $\text{Var}(aX+b) = \text{Var}(aX)$. Then, $$\begin{align} @@ -306,7 +317,7 @@ $$\begin{align} transformation

-2. Variance of sums of RVs is affected by the (in)dependence of the RVs +2. Variance of sums of RVs is affected by the (in)dependence of the RVs. $$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) 2\text{cov}(X,Y)$$ $$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \qquad \text{if } X, Y \text{ independent}$$ @@ -315,7 +326,7 @@ $$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \qquad \text{if } X, Y \text ## Proof The variance of a sum is affected by the dependence between the two random variables that are being added. Let’s expand out the definition of $\text{Var}(X + Y)$ to see what’s going on. -To simplify the math, let $\mu_x = \mathbb{E}[X]$ and $\mu_y = \mathbb{E}[Y]$ +To simplify the math, let $\mu_x = \mathbb{E}[X]$ and $\mu_y = \mathbb{E}[Y]$. $$ \begin{align} \text{Var}(X + Y) &= \mathbb{E}[(X+Y- \mathbb{E}(X+Y))^2] \\ @@ -327,7 +338,7 @@ $$ \begin{align} ::: ### Covariance and Correlation -We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to *two* random variables: $\text{Cov}(X, X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \text{Var}(X)$. +We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to *two* random variables: $\text{Cov}(X, X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \text{Var}(X)$ $$\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$$ @@ -338,7 +349,7 @@ $$r(X, Y) = \mathbb{E}\left[\left(\frac{X-\mathbb{E}[X]}{\text{SD}(X)}\right)\le It turns out we've been quietly using covariance for some time now! If $X$ and $Y$ are independent, then $\text{Cov}(X, Y) =0$ and $r(X, Y) = 0$. Note, however, that the converse is not always true: $X$ and $Y$ could have $\text{Cov}(X, Y) = r(X, Y) = 0$ but not be independent. ### Summary -* Let $X$ be a random variable with distribution $P(X=x). +* Let $X$ be a random variable with distribution $P(X=x)$. * $\mathbb{E}[X] = \sum_{x} x P(X=x)$ * $\text{Var}(X) = \mathbb{E}[(X-\mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$ * Let $a$ and $b$ be scalar values. @@ -360,10 +371,10 @@ There are several cases of random variables that appear often and have useful pr * $\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = p - p^2 = p(1-p)$ * Binomial(n, p) * Number of 1s in 'n' independent Bernoulli(p) trials. - * Let $Y$ be a Binomial(n, p) random variable - * the distribution of $Y$ is given by the binomial formula, and we can write $Y = \sum_{i=1}^n X_i$ where + * Let $Y$ be a Binomial(n, p) random variable. + * The distribution of $Y$ is given by the binomial formula, and we can write $Y = \sum_{i=1}^n X_i$ where: * $X_i$ s the indicator of success on trial i. $X_i = 1$ if trial i is a success, else 0. - * all $X_i$ are i.i.d. and Bernoulli(p) + * All $X_i$ are i.i.d. and Bernoulli(p). * $\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[X_i] = np$ * $\text{Var}(X) = \sum_{i=1}^n \text{Var}(X_i) = np(1-p)$ * $X_i$'s are independent, so $\text{Cov}(X_i, X_j) = 0$ for all i, j. @@ -409,18 +420,19 @@ Any theorem that provides the rough distribution of a statistic and doesn’t ne For a more in-depth demo check out [onlinestatbook](https://onlinestatbook.com/stat_sim/sampling_dist/). -THE CLT applies if the sample size $n$ is large, but how large does n have to be for the normal approximation to be good? It depends on the shape of the distribution of the population. +The CLT applies if the sample size $n$ is large, but how large does n have to be for the normal approximation to be good? It depends on the shape of the distribution of the population. -* If population is roughly symmetric and unimodal/uniform, could need as few as $n = 20$. -* If population is very skewed, you will need bigger n. +* If the population is roughly symmetric and unimodal/uniform, could need as few as $n = 20$. +* If the population is very skewed, you will need a bigger n. * If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped. ### Using the Sample Mean to Estimate the Population Mean Our goal with sampling is often to estimate some characteristic of a population. When we collect a single sample, it has just one average. Since our sample was random, it *could* have come out differently. The CLT helps us understand this difference. We should consider the average value and spread of all possible sample means, and what this means for how big $n$ should be. -For every sample size, the expected value of the sample mean is the population mean. $\mathbb{E}[\bar{X_n}] = \mu$. We call the sample mean an unbiased estimator of the population mean, and we'll cover this more in next lecture. +For every sample size, the expected value of the sample mean is the population mean. $\mathbb{E}[\bar{X_n}] = \mu$. We call the sample mean an unbiased estimator of the population mean, and we'll cover this more in the next lecture. + +The square root law ([Data 8](https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html#the-square-root-law)) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. $\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}$. The sample mean is more likely to be close to the population mean if we have a larger sample size. -Square root law ([Data 8](https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html#the-square-root-law)) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. $\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}$. The sample mean is more likely to be close to the population mean if we have a larger sample size.

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