Probability generating function

In summary, a probability generating function is a power series used to model the probability distribution of a discrete random variable. It differs from a moment generating function in that it is specific to discrete random variables and only generates probabilities. The main purpose of using a probability generating function is to simplify calculations and analyze random variable behavior. It is related to the moments of a distribution through its derivatives and cannot be used for continuous random variables.
  • #1
sneaky666
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Homework Statement



Let Y=x+4. Compute rY(t) in terms of rX


Homework Equations





The Attempt at a Solution



is the answer just

r 3X+4 (t)

?
 
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  • #2
What does "r" mean?
 

1. What is a probability generating function?

A probability generating function is a mathematical tool used in probability theory to model the probability distribution of a discrete random variable. It is a power series that generates the probabilities of all possible outcomes of a random variable.

2. How is a probability generating function different from a moment generating function?

A probability generating function is specific to discrete random variables, while a moment generating function can be used for both discrete and continuous random variables. Additionally, a probability generating function only generates the probabilities of outcomes, while a moment generating function generates the moments of a distribution.

3. What is the purpose of using a probability generating function?

The main purpose of using a probability generating function is to simplify calculations and make it easier to analyze the behavior of a random variable. It allows for the calculation of probabilities without having to use complex formulas or tables.

4. How is a probability generating function related to the moments of a distribution?

The probability generating function is related to the moments of a distribution through its derivatives. Specifically, the k-th derivative of the probability generating function evaluated at 1 gives the k-th moment of the distribution.

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

No, a probability generating function is only defined for discrete random variables. For continuous random variables, a moment generating function or a characteristic function is typically used instead.

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