moment generating function of the sum of independent random variables
Let Xi be independent random variables for i=1,…,n, let each Xi have moment generating function MXi(t), and let X=∑ni=1Xi. Then the moment generating function of X is
MX(t)=n∏i=1MXi(t). |
Proof.
By definition,
MX(t) | =E(etX) | ||
=E(et(X1+⋯+Xn)) | |||
=E(etX1⋯etXn). |
Now, since each Xi is independent of the others, this becomes
MX(t) | =E(etX1)⋯E(etXn) | ||
=MX1(t)⋯MXn(t) | |||
=n∏i=1MXi(t) |
as required. ∎
Title | moment generating function of the sum of independent random variables |
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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 |