10 Moment generating functions - University of California ...

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!t3X3+..]= 1 +tE(X) +12t2E(X2) +13!t3E(X3) +..The expectation of thek-th power ofX,mk=E(Xk), is called thek-thmomentofx. Incombinatorial language, then, (t) is the exponential generating function of the also thatddtE(etX)|t=0=EX,d2dt2E(etX)|t=0=EX2 ,which lets you compute the expectation and variance of a random variable once you know itsmoment generating the Moment generating function for a Poisson( ) random Moment generating FUNCTIONS120By definition, (t) = n=0etn nn!}}

10 MOMENT GENERATING FUNCTIONS 121 Why are moment generating functions useful? One reason is the computation of large devia-tions. Let Sn = X1 +···+Xn, where Xi are independent and identically distributed as X, with expectation EX= µand moment generating function φ.

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