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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)]

Q: Moment generating and the cumulant generating function of poisson distribution?
This answer is:
Related answers

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)]

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Lamdba (like most Greek letters in statistics) usually denotes a parameter of a distribution (usually of Poisson, gamma or exponential distributions). This will specify the entire distribution and allow for numerical analysis of the probability generating, moment generating, probability density/mass, distribution and/or cumulant functions (along with all moments), as and where these are defined.

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