# Probability Generating Function

• September 1st 2010, 09:31 AM
schrodingersdog
Probability Generating Function
A discrete random variable X has a probability mass function given by

px (x) = (k(p^x))/x for x=1,2...

where p is a constant such that 0 < p < 1 and k = -1/ln(q) where q = 1-p

a) Show that the probability generating function of X is given by

E(t^x) = (ln(1 - pt))/ln(q) for |t| < 1/p

b) Why must t be restricted to this range?

any help is greatly appreciated! :)
• September 1st 2010, 01:17 PM
Moo
Hello,

Well just write the expectation :

$E[t^X]=\sum_{x=1}^\infty \frac{k t^x p^x}{x}=\frac{1}{\ln(1-p)} \sum_{x=1}^\infty -\frac{(pt)^x}{x}$

but the sum is the power series of $\ln(1-pt)$, hence the result.

Why is it restricted to |t|<1/p ? Because otherwise, the series doesn't converge.