moment generating function of the sum of independent random variables


Let Xi be independent random variablesMathworldPlanetmath for i=1,,n, let each Xi have moment generating function MXi(t), and let X=i=1nXi. Then the moment generating function of X is

MX(t)=i=1nMXi(t).
Proof.

By definition,

MX(t) =E(etX)
=E(et(X1++Xn))
=E(etX1etXn).

Now, since each Xi is independent of the others, this becomes

MX(t) =E(etX1)E(etXn)
=MX1(t)MXn(t)
=i=1nMXi(t)

as required. ∎

Title moment generating function of the sum of independent random variables
Canonical name MomentGeneratingFunctionOfTheSumOfIndependentRandomVariables
Date of creation 2013-03-22 17:17:11
Last modified on 2013-03-22 17:17:11
Owner me_and (17092)
Last modified by me_and (17092)
Numerical id 5
Author me_and (17092)
Entry type Corollary
Classification msc 60E05