Value of t for Probability Generating Function

In summary, the PGF for a uniform distribution is given by ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and the condition ##G_X (1) =1## is not strictly true for all cases. The condition is actually ##G(1−) = 1, where G(1−) = limz→1G(z) from below, since the probabilities must sum to one. As for the meaning of ##G_X (2)##, it depends on the context being considered and may vary on a case-by-case basis.
  • #1
songoku
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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
 
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  • #2
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. "
 
  • #3
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
 
  • #4
It is the expectation value of 2x. Whether that is particularly meaningful is another question.
 
  • #5
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
 

1. What is the purpose of the probability generating function (PGF)?

The PGF is a mathematical tool used to describe the probability distribution of a discrete random variable. It allows us to calculate various probabilities and moments of the distribution, making it a useful tool in probability and statistics.

2. How is the value of t related to the PGF?

The value of t in the PGF represents the number of successes in a given number of trials. It is used to calculate the probabilities of different outcomes for a discrete random variable.

3. What is the significance of the value of t in the PGF?

The value of t in the PGF is important because it allows us to calculate various probabilities and moments of the distribution. It also helps us to determine the shape and characteristics of the distribution, such as its mean and variance.

4. How is the value of t determined for a specific distribution?

The value of t for a specific distribution is determined based on the number of trials and the probability of success for each trial. It can be calculated using the formula t = (1-p)/p, where p is the probability of success.

5. Can the value of t be negative in the PGF?

No, the value of t in the PGF cannot be negative. It represents the number of successes, which must be a non-negative integer. If the value of t is negative, it would not make sense in the context of the distribution and the probabilities calculated using the PGF would not be valid.

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