# HG changeset patch
# User Oleksandr Gavenko <gavenkoa@gmail.com>
# Date 1458129604 -7200
# Node ID 5d6cec5fe095f5c289fc193d37cfa8d8fb7fdd33
# Parent  3b9b5823948c64d49597b49de79ec8a0730a7739
Probability density function. Continuous uniform random variable. Exponential
random variables. Cumulative distribution functions.

diff -r 3b9b5823948c -r 5d6cec5fe095 probability-continuous.rst
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/probability-continuous.rst	Wed Mar 16 14:00:04 2016 +0200
@@ -0,0 +1,158 @@
+
+=============================
+ Continuous random variables
+=============================
+.. contents::
+   :local:
+
+Probability density function
+============================
+
+.. role:: def
+   :class: def
+
+:def:`Probability density function` (PDF) for continuous random variable
+:math:`x` is function:
+
+.. math::
+
+   PDF(a ≤ X ≤ b) = P(a ≤ X ≤ b) = ∫_{a, b} f_X(x) dx
+
+   f_X(x) ≥ 0
+
+   ∫_{-∞, +∞} f_X(x) dx = 1
+
+:math:`f_X(x)` funtion maps values :math:`x` from sample space to real numbers.
+
+For continuous random variable:
+
+.. math:: P(X = a) = 0
+
+Expectation
+===========
+
+:def:`Expectation` of continuous random variable is:
+
+.. math:: μ = E[X] = ∫_{-∞, +∞} x·f_X(x)·dx
+
+Properties:
+
+.. math::
+
+   E[X + Y] = E[X] + E[Y]
+
+   E[a·X] = a·E[X]
+
+   E[a·X + b] = a·E[X] + b
+
+Variance
+========
+
+:def:`Variance` of continuous random variable is:
+
+.. math:: var[X] = ∫_{-∞, +∞} (x-μ)²·f_X(x)·dx
+
+Properties:
+
+.. math::
+
+   var[a·X + b] = a²·var[X]
+
+   var[X] = E[X²] - E²[X]
+
+Standard deviation
+==================
+
+:def:`Standard deviation` of continuous random variable is:
+
+.. math:: σ_Χ = sqrt(var[X])
+
+
+Continuous uniform random variable
+==================================
+
+:def:`Continuous uniform random variable` is :math:`f_X(x)` that is non-zero
+only on :math:`[a, b]` with :math:`f_X(x) = `1/(b-a)`.
+
+.. math::
+
+   E[unif(a, b)] = (b+a)/2
+
+   var[unif(a, b)] = (b-a)²/12
+
+   σ = (b-a)/sqrt(12)
+
+Proofs:
+
+.. math::
+
+   E[unif(a, b)] = ∫_{a, b} x·1/(b-a)·dx = x²/2/(b-a) |_{a, b} = (b²-a²)/(b-a)/2 = (b+a)/2
+
+   E[unif²(a, b)] = ∫_{a, b} x²·1/(b-a)·dx = x³/3/(b-a) |_{a, b} = (b³-a³)/(b-a)/3 = (b²+b·a+a²)/3
+
+   var[unif(a, b)] = E[unif²(a, b)] - E²[unif(a, b)] = (b²+b·a+a²)/3 - (b+a)²/4 = (b-a)²/12
+
+.. note::
+
+   In maxima::
+
+     (%i4) factor((b^2+b*a+a^2)/3 - (a+b)^2/4);
+                 2
+          (b - a)
+          --------
+             12
+
+Exponential random variables
+============================
+
+:def:`Exponential random variables` with parameter :math:`λ` is:
+
+.. math:: f_X(x) = λ·exp(-λ·x)
+
+for :math:`x ≥ 0`, and zero otherwise.
+
+Properties:
+
+.. math::
+
+   E[exp(λ)] = 1/λ
+
+   var[exp(λ)] = 1/λ²
+
+Proof:
+
+.. math::
+
+  ∫_{-∞, +∞} f_X(x)·dx = ∫_{0, +∞} λ·exp(-λ·x)·dx = -exp(-λ·x) |_{0, +∞} = 1
+
+  E[exp(λ)] = ∫_{0, +∞} x·λ·exp(-λ·x)·dx = 1/λ
+
+  E[exp²(λ)] = ∫_{0, +∞} x²·λ·exp(-λ·x)·dx = 1/λ²
+
+.. note::
+
+   From maxima::
+
+    (%i15) assume(lambda>0);
+    (%o15)                           [lambda > 0]
+
+    (%i16) integrate(lambda*%e^(-lambda*x),x,0,inf);
+    (%o16)                                 1
+
+    (%i17) integrate(x*lambda*%e^(-lambda*x),x,0,inf);
+                                          1
+    (%o17)                              ------
+                                        lambda
+
+    (%i18) integrate(x^2*lambda*%e^(-lambda*x),x,0,inf);
+                                           2
+    (%o18)                              -------
+                                              2
+                                        lambda
+
+Cumulative distribution functions
+=================================
+
+:def:`Cumulative distribution functions` of random variable :math:`X` is:
+
+.. math:: F_X(x) = P(X ≤ x) = ∫_{-∞, x} f_X(t)·dt