# HG changeset patch
# User Oleksandr Gavenko <gavenkoa@gmail.com>
# Date 1461309955 -10800
# Node ID c18d218b854e465975215a85d00c65cf971410a9
# Parent  db3d7a44583b78a1d040007a11afbb201050ff4b
Sum of random number of i.i.r.v.

diff -r db3d7a44583b -r c18d218b854e probability-continuous.rst
--- a/probability-continuous.rst	Thu Apr 21 16:22:14 2016 +0300
+++ b/probability-continuous.rst	Fri Apr 22 10:25:55 2016 +0300
@@ -469,3 +469,27 @@
 
 https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
 
+Sum of random number of i.i.r.v.
+================================
+
+Let :math:`Y = X_1+...+X_N` is a sum of r.v. :math:`N` and all :math:`X_i` are
+i.i.r.v. Thus:
+
+.. math:: E[Y] = E[N]·E[X]
+
+.. math:: var[Y] = E[N]·var(X) + var(N)·(E[N])²
+
+Proofs:
+
+.. math::
+
+   E[Y] = E[∑_{i=1..N}\ X_i] = E[E[∑_{i=1..n}\ X_i |N=n]] = E[∑_{i=1..n}\ E[X_i|N=n]]
+
+   = E[∑_{i=1..n}\ E[X]] = E[N·E[X]] = E[N]·E[X]
+
+.. math::
+
+   var[Y] = E[var(Y|N)] + var(E[Y|N]) = E[var(∑_{i=1..n}\ X_i |N=n)] + var(E[∑_{i=1..n}\ X_i |N=n])
+
+   = E[N·var(X)] + var(N·E[X]) = E[N]·var(X) + var(N)·(E[X])²
+