# Probability Generating Functions

• EthanW
In summary, the conversation is about the speaker trying to understand Probability Generating Functions and asking for help with solving the PGF of a random variable Y. They provide an example of a PGF for a random variable X and express confusion about how to proceed with finding the PGF for Y. The expert summarizer then explains that the original PGF indicates equal probabilities for three outcomes, and provides a formula for a different PGF that would have equal probabilities for three different outcomes. The speaker expresses gratitude for the information and mentions that they now have a better understanding of the concept.
EthanW
Hello,

I am trying to get the hang of Probability Generating Functions, but I don't quite understand them fully.

For example, I've got the PGF of a random variable X, called H:
$$H(s) = \frac{1}{3}\cdot(1+s+s^2)$$

Now, then there is a random variable Y, with Y = X + 1, and I want to solve the PGF of Y I do:
$$G_Y(s) = G_{x+1}(s) = E[s^{x+1}] = E[s^x]\cdot E = ?$$

I don't know how to go further at this point, can someone point me in the right direction?

Thanks.

EthanW said:
Hello,

I am trying to get the hang of Probability Generating Functions, but I don't quite understand them fully.

For example, I've got the PGF of a random variable X, called H:
$$H(s) = \frac{1}{3}\cdot(1+s+s^2)$$

Now, then there is a random variable Y, with Y = X + 1, and I want to solve the PGF of Y I do:
$$G_Y(s) = G_{x+1}(s) = E[s^{x+1}] = E[s^x]\cdot E = ?$$

I don't know how to go further at this point, can someone point me in the right direction?

Thanks.

Your original PGF simply means that P(0) = P(1) = P(2) = 1/3.

If you want P(1) = P(2) = P(3) = 1/3, the generating function is

$$\frac{1}{3} (s + s^2 + s^3)$$

Thanks, very useful information. I made some other exercises and they've become more clear now. :)

## What is a probability generating function?

A probability generating function (PGF) is a mathematical tool used in probability theory to describe the distribution of a discrete random variable. It is a power series representation of the probability mass function of the random variable.

## How is a probability generating function calculated?

A probability generating function is calculated by taking the sum of all possible outcomes of a random variable, multiplied by their respective probabilities, and raising the result to a power equal to the number of outcomes. This can also be expressed as the expected value of the random variable raised to the power of the variable itself.

## What is the purpose of a probability generating function?

The main purpose of a probability generating function is to simplify the computation of moments of a random variable. It can also be used to derive other useful properties of the distribution, such as the mean, variance, and higher order moments.

## How is a probability generating function used in probability distributions?

A probability generating function is commonly used in the study of discrete probability distributions, such as the binomial, Poisson, and negative binomial distributions. It allows for a more efficient and accurate way of determining the probability of certain outcomes and calculating moments of the distribution.

## Can a probability generating function be used for continuous random variables?

No, a probability generating function is only applicable to discrete random variables. For continuous random variables, other tools such as the moment generating function or characteristic function are used to describe the distribution.

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