Value of t for Probability Generating Function

[songoku]

TL;DR

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

 

[mjc123]

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

 

[songoku]

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

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

Thanks

 

[mjc123]

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

 

[songoku]

It is the expectation value of 2^x. 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