09-Distributions.Rmdhidden

# Important distributions

These distributions are used quite frequently in Bayesian data analyses, especially in psychology and linguistics applications. The Binomial and Poisson are discrete distributions, the rest are continuous. Each distribution comes with a family of ```d-p-q-r``` functions in R which allow us to compute the PDF/PMF, the CDF, the inverse CDF, and to generate random data. For example, the normal distribution's PDF is ```dnorm```; the CDF and the inverse CDF are ```pnorm``` and ```qnorm``` respectively; and random data can be generated using ```rnorm```.

The table below is adapted from https://github.com/wzchen/probability_cheatsheet, which is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.



to-do: check that the notation is consistent with the main text's.

\small
\begin{center}
\renewcommand{\arraystretch}{3.7}
\begin{tabular}{ccccc}
\textbf{Distribution} & \textbf{PMF/PDF and Support} & \textbf{Expected Value}  & \textbf{Variance}\\
\hline 
\shortstack{Binomial \\ $Binomial(n, \theta)$} & \shortstack{$P(X=k) = \binom{n}{k}\theta^k (1-\theta)^{n-k}$  \\ $k \in \{0, 1, 2, \dots n\}$}& $n\theta$ & $n\theta(1-\theta)$ \\
\hline
\shortstack{Poisson \\ $Pois(\lambda)$} & \shortstack{$P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}$ \\ $k \in \{$0, 1, 2, \dots $\}$} & $\lambda$ & $\lambda$ \\
\hline
\shortstack{Uniform \\ $Unif(a, b)$} & \shortstack{$ f(x) = \frac{1}{b-a}$ \\$ x \in (a, b) $} & $\frac{a+b}{2}$ & $\frac{(b-a)^2}{12}$ \\
\hline
\shortstack{Normal \\ $Normal(\mu, \sigma)$} & \shortstack{$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{(2 \sigma^2)}}$ \\ $x \in (-\infty, \infty)$} & $\mu =  \frac{\sum_{i=1}^n x_i}{n}$ & $\sigma^2 = \frac{\sum_{i=1}^n (x_i-\bar{x})^2}{n}$ \\
\hline
\shortstack{Log-Normal \\ $LogNormal(\mu,\sigma)$} & \shortstack{$\frac{1}{x\sigma \sqrt{2\pi}}e^{-(\log x - \mu)^2/(2\sigma^2)}$\\$x \in (0, \infty)$} & $\theta = e^{ \mu + \sigma^2/2}$ & $\theta^2 (e^{\sigma^2} - 1)$ \\
\hline
\shortstack{Beta \\ Beta($a, b$)} & \shortstack{$f(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}$\\$x \in (0, 1) $} & $\mu = \frac{a}{a + b}$  & $\frac{\mu(1-\mu)}{(a + b + 1)}$  \\
\hline
\shortstack{Exponential \\ $Exp(\lambda)$} & \shortstack{$f(x) = \lambda e^{-\lambda x}$\\$ x \in (0, \infty)$} & $\frac{1}{\lambda}$  & $\frac{1}{\lambda^2}$ \\
\hline
\shortstack{Gamma \\ $Gamma(a, \lambda)$} & \shortstack{$f(x) = \frac{1}{\Gamma(a)}(\lambda x)^ae^{-\lambda x}\frac{1}{x}$\\$ x \in (0, \infty)$} & $\frac{a}{\lambda}$  & $\frac{a}{\lambda^2}$ \\
\hline
\shortstack{Student-$t$ \\ $t(n)$ \\ Cauchy is $t(1)$} & \shortstack{$\frac{\Gamma((n+1)/2)}{\sqrt{n\pi} \Gamma(n/2)} (1+x^2/n)^{-(n+1)/2}$\\$x \in (-\infty, \infty)$} & $0$ if $n>1$ & $\frac{n}{n-2}$ if $n>2$ \\
\hline
\end{tabular}
\end{center}