# HG changeset patch
# User Oleksandr Gavenko <gavenkoa@gmail.com>
# Date 1459419963 -10800
# Node ID 5a09c6837dcbfeb126cc75b2e99043ca5c9aa101
# Parent  d4bed13a2c3702d6f3c9a08f3374fc6b41e5cede
Covariance

diff -r d4bed13a2c37 -r 5a09c6837dcb probability-continuous.rst
--- a/probability-continuous.rst	Thu Mar 31 00:15:20 2016 +0300
+++ b/probability-continuous.rst	Thu Mar 31 13:26:03 2016 +0300
@@ -293,14 +293,14 @@
 
 and so:
 
-.. math:: f_Y(y) = (d\ f_Y(t)/dt)(y) = (d\ F_X(h(t))/dt)(y) = F_X(h(y))·(d\ h(t)/dt)(y)
+.. math:: f_Y(y) = (d\ F_Y(t)/dt)(y) = (d\ F_X(h(t))/dt)(y) = f_X(h(y))·(d\ h(t)/dt)(y)
 
 Convolution formula
 ===================
 
 If :math:`Z = X + Y` and X and Y is independent r.v. then:
 
-.. math:: f_Z(z) = ∫_x\ f_X(x)·f_Y(z-x)̣·dx
+.. math:: f_Z(z) = ∫_x\ f_X(x)·f_Y(z-x)·dx
 
 Proof:
 
@@ -320,3 +320,50 @@
 
 .. math:: f_Z(z) = ∫_x\ f_{X,Z}(x,z)\ dx = ∫_x\ f_X(x)·f_Y(z-x)\ dx
 
+Covariance
+==========
+
+Covariance of two r.v. is:
+
+.. math:: cov(X, Y) = E[(X - E[X])·(Y - E[Y])]
+
+Properties:
+
+.. math:: cov(X, Y) = E[X·Y] - E[X]·E[Y]
+
+.. math:: cov(X, X) = var(X)
+
+.. math:: cov(a·X + b, Y) = a·cov(X, Y)
+
+.. math:: cov(X, Y + Z) = cov(X, Y) + cov(X, Z)
+
+.. math:: var(X + Y) = var(X) + var(Y) + 2·cov(X, Y)
+
+Covariance of two independent r.v. is zero.
+
+Proof:
+
+.. math::
+
+   cov(X, Y) = E[(X - E[X])·(Y - E[Y])] = E[(X - E[X])]·E[(Y - E[Y])] = 0
+
+Correlation coefficient
+=======================
+
+Dimensionless version of covariance:
+
+.. math:: ρ(Χ, Υ) = E[(X-E[X])/σ_Χ·(Y-E[Y])/σ_Y] = cov(X, Y)/(σ_X·σ_Y)
+
+It is defined only for cases when :math:`σ_X ≠ 0` and :math:`σ_Y ≠ 0`.
+
+Obviously :math:`-1 ≤ ρ(Χ, Υ) ≤ +1` and :math:`ρ(Χ, X) = 0`.
+
+For independent r.v. :math:`ρ(Χ, Y) = 0`.
+
+If :math:`|ρ(X, Y)| = 1` then :math:`X` and :math:`Y` is have linear
+dependencies :math:`X = Y` or :math:`X = -Y`.
+
+Properties:
+
+.. math:: ρ(a·X + b, Y) = sign(a)·ρ(X, Y)
+