Introduction to Probability · Problem 10.1.9

Generating Functions — Grinstead & Snell

Mathematics — Probability · generating-functions-discrete

Problem

Let

X

and

Y

be random variables with values in

\{1, 2, 3, 4, 5, 6\}

with distribution functions

p_X

and

p_Y

given by

p_X(j) = a_j, \quad p_Y(j) = b_j.

(a) Find the ordinary generating functions

h_X(z)

and

h_Y(z)

for these distributions. (b) Find the ordinary generating function

h_Z(z)

for the distribution

Z = X + Y

. (c) Show that

h_Z(z)

cannot ever have the form

h_Z(z) = \frac{z^2 + z^3 + \cdots + z^{12}}{11}.

Hint:

h_X

and

h_Y

must have at least one nonzero root, but

h_Z(z)

in the form given has no nonzero real roots. It follows from this observation that there is no way to load two dice so that the probability that a given sum will turn up when they are tossed is the same for all sums (i.e., that all outcomes are equally likely).

Related theory

Characteristic Functions And Moment Generating Functions

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