472 Sum of random number of i.i.r.v.

   473 ================================

   474 

   475 Let :math:`Y = X_1+...+X_N` is a sum of r.v. :math:`N` and all :math:`X_i` are

   476 i.i.r.v. Thus:

   477 

   478 .. math:: E[Y] = E[N]·E[X]

   479 

   480 .. math:: var[Y] = E[N]·var(X) + var(N)·(E[N])²

   481 

   482 Proofs:

   483 

   484 .. math::

   485 

   486    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]]

   487 

   488    = E[∑_{i=1..n}\ E[X]] = E[N·E[X]] = E[N]·E[X]

   489 

   490 .. math::

   491 

   492    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])

   493 

   494    = E[N·var(X)] + var(N·E[X]) = E[N]·var(X) + var(N)·(E[X])²

   495