# HG changeset patch
# User Oleksandr Gavenko <gavenkoa@gmail.com>
# Date 1458245091 -7200
# Node ID 9b4e31a031618031b469b8c9e323ddc57ae0b28e
# Parent  60353d4d994d619514da29c17ef01b888fd8bca6
Normal random variables.

diff -r 60353d4d994d -r 9b4e31a03161 probability-continuous.rst
--- a/probability-continuous.rst	Wed Mar 16 16:18:11 2016 +0200
+++ b/probability-continuous.rst	Thu Mar 17 22:04:51 2016 +0200
@@ -16,11 +16,11 @@
 
 .. math::
 
-   PDF(a ≤ X ≤ b) = P(a ≤ X ≤ b) = ∫_{a, b} f_X(x) dx
+   PDF(a ≤ X ≤ b) = P(a ≤ X ≤ b) = ∫_{a, b}\ f_X(x) \ dx
 
    f_X(x) ≥ 0
 
-   ∫_{-∞, +∞} f_X(x) dx = 1
+   ∫_{-∞, +∞}\ f_X(x) \ dx = 1
 
 :math:`f_X(x)` funtion maps values :math:`x` from sample space to real numbers.
 
@@ -33,7 +33,7 @@
 
 :def:`Expectation` of continuous random variable is:
 
-.. math:: μ = E[X] = ∫_{-∞, +∞} x·f_X(x)·dx
+.. math:: μ = E[X] = ∫_{-∞, +∞}\ x·f_X(x) \ dx
 
 Properties:
 
@@ -50,7 +50,7 @@
 
 :def:`Variance` of continuous random variable is:
 
-.. math:: var[X] = ∫_{-∞, +∞} (x-μ)²·f_X(x)·dx
+.. math:: var[X] = ∫_{-∞, +∞}\ (x-μ)²·f_X(x) \ dx
 
 Properties:
 
@@ -67,6 +67,52 @@
 
 .. math:: σ_Χ = sqrt(var[X])
 
+Cumulative distribution functions
+=================================
+
+:def:`Cumulative distribution functions` (CDF) of random variable :math:`X` is:
+
+.. math:: F_X(x) = P(X ≤ x) = ∫_{-∞, x}\ f_X(t) \ dt
+
+So:
+
+.. math::
+
+   P(a ≤ X ≤ b) = F_X(b) - F_X(a) + f_X(a) = ∫_{a,b}\ f_X(x) \ dx
+
+   F_X(-∞) = 0
+
+   F_X(+∞) = 1
+
+and :math:`F_X(a) ≤ F_X(b)` for :math:`a ≤ b`.
+
+Relation between CDF and PDF:
+
+.. math:: (d(CDF(t))/dt)(x) = PDF(x)
+
+Conditional probability
+=======================
+
+:def:`Conditional probability` of continuous random variable is:
+
+.. math:: P(X ⊆ B | A) = ∫_{B}\ f_{X|A}(x) \ dx = ∫_{A∩B}\ f_X(x) \ dx / P(A)
+
+:def:`Conditional expectation` of continuous random variable is:
+
+.. math:: E[X|A] = ∫_\ x·f_{X|A}(x) \ dx
+
+Properties:
+
+.. math::
+
+   E[g(X)|A] = ∫_\ g(x)·f_{X|A}(x) \ dx
+
+Independence
+============
+
+Random variable :math:`X`, :math:`Y` are :def:`independent` if:
+
+.. math:: f_{X,Y}(x, y) = f_X(x)·f_Y(y)
 
 Continuous uniform random variable
 ==================================
@@ -86,7 +132,7 @@
 
 .. 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[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
 
@@ -123,11 +169,11 @@
 
 .. math::
 
-  ∫_{-∞, +∞} f_X(x)·dx = ∫_{0, +∞} λ·exp(-λ·x)·dx = -exp(-λ·x) |_{0, +∞} = 1
+  ∫_{-∞, +∞}\ 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[exp(λ)] = ∫_{0, +∞}\ x·λ·exp(-λ·x) \ dx = 1/λ
 
-  E[exp²(λ)] = ∫_{0, +∞} x²·λ·exp(-λ·x)·dx = 1/λ²
+  E[exp²(λ)] = ∫_{0, +∞}\ x²·λ·exp(-λ·x) \ dx = 1/λ²
 
 .. note::
 
@@ -150,9 +196,28 @@
                                               2
                                         lambda
 
-Cumulative distribution functions
-=================================
+Normal random variables
+=======================
+
+:def:`Normal random variables` with parameters :math:`μ, σ` and :math:`σ > 0`
+defined by PDF:
+
+.. math:: norm(μ, σ) = 1/σ/sqrt(2·π)·exp(-(x-μ)²/σ²/2)
+
+Properties:
+
+.. math::
 
-:def:`Cumulative distribution functions` of random variable :math:`X` is:
+   E[norm(μ, σ)] = μ
+
+   var[norm(μ, σ)] = σ²
+
+Disjoint distribution of two normal r.v.
+========================================
 
-.. math:: F_X(x) = P(X ≤ x) = ∫_{-∞, x} f_X(t)·dt
+.. math::
+
+   norm2(μ₁, μ₂, σ₁, σ₂) = norm(μ₁, σ₁)·norm(μ₂, σ₂)
+
+   = 1/(2·π·σ₁·σ₂)·exp(-(x-μ₁)²/σ₁²/2 - (x-μ₂)²/σ₂²/2)
+