Grinstead-Snell-Probability · Problem 10.1.6

Generating Functions

Mathematics · generating-functions-discrete

Problem

Let

p

be the probability distribution

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

and let

p_n = p * p * \cdots * p

be the

n

-fold convolution of

p

with itself. (a) Find

p_2

by direct calculation (see Definition 7.1). (b) Find the ordinary generating functions

h(z)

and

h_2(z)

for

p

and

p_2

, and verify that

h_2(z) = (h(z))^2

. (c) Find

h_n(z)

from

h(z)

. (d) Find the first two moments, and hence the mean and variance, of

p_n

from

h_n(z)

. Verify that the mean of

p_n

n

times the mean of

p

. (e) Find those integers

j

for which

p_n(j) > 0

from

h_n(z)