=+27. Let X be a random variable with moment generating function M(t)=E etX = n=0 n n! t n 100 4. Combinatorics defined in

=+27. Let X be a random variable with moment generating function M(t)=E etX = ∞

n=0

μn n!

t n

100 4. Combinatorics defined in some neighborhood of the origin. Here μn = E(Xn) is the nth moment of X. The function ln M(t) = ∞

n=1

κn n!

t n

is called the cumulant generating function, and its nth coefficient κn is called the nth cumulant. Based on Fa`a di Bruno’s formula, show that

μn =  n!

n m=1 bm!(m!)bm

κb1 1 ··· κbn n

κn =  n!(−1)k−1(k − 1)!

n m=1 bm!(m!)bm

μb1 1 ··· μbn n ,