Moment generating and the cumulant generating function of poisson distribution?

Answer 1

The moment generating function is

M(t) = Expected value of e^(xt)

= SUM[e^(xt)f(x)]

and for the Poisson distribution with mean a

inf

= SUM[e^(xt).a^x.e^(-a)/x!]

x=0

inf

= e^(-a).SUM[(ae^t)^x/x!]

x=0

= e^(-a).e^(ae^t)

= e^[a(e^t -1)]