# Category:Probability Generating Functions

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This category contains results about Probability Generating Functions.

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

The **probability generating function for $X$**, denoted $\map {\Pi_X} s$, is the formal power series defined by:

- $\ds \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \sqbrk {\sqbrk s}$

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Probability Generating Functions"

The following 24 pages are in this category, out of 24 total.

### E

### P

- PGF of Sum of Independent Discrete Random Variables
- PGF of Sum of Independent Discrete Random Variables/General Result
- PGF of Sum of Random Number of Discrete Random Variables
- Probability Generating Function as Expectation
- Probability Generating Function defines Probability Distribution
- Probability Generating Function of Bernoulli Distribution
- Probability Generating Function of Binomial Distribution
- Probability Generating Function of Degenerate Distribution
- Probability Generating Function of Discrete Uniform Distribution
- Probability Generating Function of Geometric Distribution
- Probability Generating Function of Negative Binomial Distribution
- Probability Generating Function of One
- Probability Generating Function of Poisson Distribution
- Probability Generating Function of Scalar Multiple of Random Variable
- Probability Generating Function of Shifted Geometric Distribution
- Probability Generating Function of Shifted Random Variable
- Probability Generating Function of Zero
- Properties of Probability Generating Function