Fix Histogram distribution text and E(x) formula

mef/stochastic_layer.rst

@@ -554,7 +554,7 @@ As for arithmetic operators and built-ins, this list can be extended on demand.
     +-----------------------+------------+-------------------------------------------------------------------------------------------------------------+
     | **beta-deviate**      | 2          | beta distributions defined by two shape parameters :math:`\alpha` and :math:`\beta`                         |
     +-----------------------+------------+-------------------------------------------------------------------------------------------------------------+
-    | **histograms**        | any        | discrete distributions defined by means of a list of pairs                                                  |
+    | **histogram**         | >1         | piecewise-constant distributions defined by means of a list of pairs                                        |
     +-----------------------+------------+-------------------------------------------------------------------------------------------------------------+
 
 Uniform Deviates
@@ -665,53 +665,62 @@ Beta Deviates
     The default value of the beta distribution is its mean, i.e., :math:`\alpha/(\alpha + \beta)`.
 
 Histograms
-    Histograms are lists of pairs :math:`(x_1, E_1), \ldots, (x_n, E_n)`,
-    where the :math:`x_i`'s are numbers
-    such that :math:`x_i < x_{i+1} \text{ for } i=1, \ldots, n-1`
-    and the :math:`E_i`'s are expressions.
+    Histograms are lists of pairs :math:`(b_1, w_1), \ldots, (b_n, w_n)`,
+    where the :math:`b_i`'s are upper bounds of successive, contiguous intervals
+    such that :math:`b_i < b_{i+1} \text{ for } i=0, \ldots, n-1`,
+    and the :math:`w_i`'s are non-negative weights for the intervals :math:`[b_{i-1}, b_i)`.
+    The lower bound of the first interval :math:`b_0` is given apart.
 
-    The :math:`x_i`'s represent upper bounds of successive intervals.
-    The lower bound of the first interval :math:`x_0` is given apart.
+    The drawing of a value according to a histogram distribution is a two-step process.
+    First, the interval :math:`i` is drawn at random
+    from a discrete distribution with the corresponding weights :math:`w_i`'s;
+    then, a random value :math:`x` is drawn uniformly from the range :math:`[b_{i-1}, b_i)`.
+    The sampling of the intervals and random values must be independent.
 
-    The drawing of a value according to a histogram is a two-step process.
-    First, a value :math:`z` is drawn uniformly in the range :math:`[x_0, x_n]`.
-    Then, a value is drawn at random by means of the expression :math:`E_i`,
-    where :math:`i` is the index of the interval
-    such that :math:`x_{i-1} < z \leq x_i`.
+    The probability density function of the histogram (or piece-wise constant) distribution:
 
-    By default, the value of a histogram is its mean, i.e.,
+    .. math::
+
+        f(x;b_0,\ldots,b_n, w_1,\ldots,w_n) = \dfrac{w_k}{(b_k - b_{k-1})\cdot\sum_{i=1}^{n}w_i}
+
+    Where :math:`k` is such that
+
+    .. math::
+
+        b_{k - 1} \leq x < b_k \quad \forall k \in \mathbb{Z} : 1 \leq k \leq n
+
+    By default, the value of the histogram distribution is its mean, i.e.,
 
     .. math::
 
-        \mathbf{E}(X) = \frac{1}{x_n - x_0} \times \sum_{i=1}^{n}(x_i - x_{i-1})\mathbf{E}(E_i)
+        E(x) = \dfrac{\sum_{i=1}^{n}\tfrac{1}{2}(b_i + b_{i-1}) \cdot w_i}{\sum_{i=1}^{n}w_i}
 
     Both Cumulative Distribution Functions
-    and Density Probability Distributions can be translated into histograms.
+    and Discrete Probability Distributions can be translated into histograms.
 
     A Cumulative Distribution Function is a list of pairs
-    :math:`(p_1, v_1), \ldots, (p_n, v_n)`,
+    :math:`(p_1, b_1), \ldots, (p_n, b_n)`,
     where the :math:`p_i`'s are
-    such that :math:`p_i < p_{i+1} \text{ for } i=1, \ldots, n \text{ and } p_n=1`.
+    such that :math:`p_i < p_{i+1} \text{ for } i=0, \ldots, n-1 \text{ and } p_n=1, p_0=0`.
     It differs from histograms in two ways.
-    First, :math:`X` axis values are normalized (to spread between 0 and 1);
+    First, :math:`Y` axis values are normalized (to spread between 0 and 1);
     second, they are presented in a cumulative way.
     The histogram that corresponds to a Cumulative Distribution Function
-    :math:`(p_1, v_1), \ldots, (p_n, v_n)`
-    is the list of pairs :math:`(x_1, v_1), \ldots, (x_n, v_n)`,
-    with the initial value
-    :math:`x_0 = 0, x_1 = p_1, \text{ and } x_i = p_i - p_{i-1} \text{ for all } i>1`.
+    :math:`(p_1, b_1), \ldots, (p_n, b_n)`
+    is the list of pairs :math:`(b_1, w_1), \ldots, (b_n, w_n)`,
+    where :math:`w_i = p_i - p_{i-1}`.
 
     A Discrete Probability Distribution is a list of pairs
     :math:`(d_1, m_1), \ldots, (d_n, m_n)`.
     The :math:`d_i`'s are probability densities.
-    However, they could be any kind of values.
+    However, they could be any kind of non-negative values.
     The :math:`m_i`'s are midpoints of intervals
-    and are such that :math:`m_1 < m_2 < \ldots < m_n < 1`.
+    and are such that :math:`0 < m_1 < m_2 < \ldots < m_n`.
     The histogram that corresponds to a Discrete Probability Distribution
     :math:`(d_1, m_1), \ldots, (d_n, m_n)`
-    is the list of pairs :math:`(x_1, d_1), \ldots, (x_n, d_n)`,
-    with the initial value
-    :math:`x_0 = 0, x_1 = 2m_1, \text{ and } x_i = x_{i-1} + 2(m_i - x_{i-1})`.
+    is the list of pairs :math:`(b_1, d_1), \ldots, (b_n, d_n)`,
+    with the initial boundary :math:`b_0 = 0`,
+    :math:`b_i = b_{i-1} + 2(m_i - b_{i-1})`.
 
 
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