Transcription of 10 Moment generating functions - Mathematics Home
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10 Moment generating functions - Mathematics Home
10 Moment generating FUNCTIONS11910 Moment generating functionsIfXis a random variable, then itsmoment generating functionis (t) = X(t) =E(etX) ={ xetxP(X=x) in discrete case, etxfX(x)dxin continuous thatXis Exponential(1) random variable, that is,fX(x) ={e xx >0,0x , (t) = 0etxe xdx=11 t,only whent <1. Otherwise the integral diverges and the Moment generating function does notexist. Have in mind that Moment generating function is only meaningful when the integral (orthe sum) is where the name comes from. Writing its Taylor expansion in place ofetXandexchanging the sum and the integral (which can be done in manycases)E(etX) =E[1 +tX+12t2X2+13!]}}
Compute the moment generating function of a uniform random variable on [0,1]. 3. This exercise was in fact the original motivation for the study of large deviations, by the Swedish probabilist Harald Cram`er, who was working as an insurance company consultant in 1930’s. Assume that the insurance company receives a steady stream of payments ...
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