# Value of t for Probability Generating Function

• B
• songoku

#### songoku

TL;DR Summary
Let PGF be

$$G_X (t) = E(t^x) = \Sigma P(X=x_i) t^{x_i}$$

and ##G_X (1) = 1##
My questions:

1) What about if t = 2? Is there a certain meaning to ##G_X (2)## ?

2) PGF for uniform distribution is ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and for t = 1 ##G_X (1)## is undefined so ##G_X (1) =1## is not true for all cases?

Thanks

GX(1) = 1 is not strictly correct. The condition is (to quote Wikipedia)
" G(1−) = 1, where G(1−) = limz→1G(z) from below, since the probabilities must sum to one. "

GX(1) = 1 is not strictly correct. The condition is (to quote Wikipedia)
" G(1−) = 1, where G(1−) = limz→1G(z) from below, since the probabilities must sum to one. "

What about ##G_X (2)## ? Is there a certain meaning to it?

Thanks

It is the expectation value of 2x. Whether that is particularly meaningful is another question.

It is the expectation value of 2x. Whether that is particularly meaningful is another question.
So whether it is meaningful or not depend on the context being considered so it will be more like case-by-case basis?

Thanks