B Value of t for Probability Generating Function

AI Thread Summary
The discussion focuses on the implications of the probability generating function (PGF) when t equals 2, indicating that G_X(2) represents the expectation value of 2x. It clarifies that for uniform distributions, G_X(1) is not universally defined as 1, but rather G(1−) approaches 1, reflecting the requirement that probabilities sum to one. The participants note that the significance of G_X(2) can vary depending on the specific context in which it is applied. Overall, the conversation emphasizes the nuanced understanding of PGFs in probability theory. The importance of context in interpreting these values is highlighted throughout the discussion.
songoku
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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
 
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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. "
 
mjc123 said:
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.
 
mjc123 said:
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
 
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