Question: Let X be a scalar random variable whose distribution is Poisson with mean 1. Show that the cumulants of X of all orders are equal

Let X be a scalar random variable whose distribution is Poisson with mean 1.

Show that the cumulants of X of all orders are equal to 1. Hence show that the rth moment is

μ

′

r = E (Xr) = Br where Br

, the rth Bell number, is the number of partitions of a set of r elements.

Hence derive a generating function for the Bell numbers.

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