Introduction to Probability · Problem 10.1.5

Generating Functions — Grinstead & Snell

Mathematics — Probability · generating-functions-discrete

Problem

Let

p

and

p'

be the two distributions

p = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1/3 & 0 & 0 & 2/3 & 0 \end{pmatrix},

p' = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 2/3 & 0 & 0 & 1/3 \end{pmatrix}.

(a) Show that

p

and

p'

have the same first and second moments, but not the same third and fourth moments. (b) Find the ordinary and moment generating functions for

p

and

p'