3. Probability Generating Function (PGF)

Concept

The PGF is a power series representation of the probability distribution of a discrete random variable. It converts a probability problem into an algebraic problem.

Definition

For a discrete random variable $X$ taking non-negative integer values: $$ G_X(t) = E[t^X] = \sum_{x=0}^{\infty} t^x P(X=x) $$

How to use PGF to find Moments

1. Probabilities: Differentiate $G_X(t)$ once.

$$ P(X=x) = \frac{1}{x!} \left[ \frac{d^x G_X(t)}{dt^x} \right]_{t=0} $$

2. Mean: Differentiate once and set $t=1$.

$$ E[X] = G'_X(1) $$

3. Variance: Requires second derivative.

$$ Var(X) = G''_X(1) + G'_X(1) - [G'_X(1)]^2 $$

Detailed Example

Problem: Find the PGF, Mean, and Variance for a Bernoulli distribution (Success/Failure) where $P(1)=p$ and $P(0)=q$.

Solution:

1. Construct PGF:

$$ G_X(t) = t^0 P(0) + t^1 P(1) = q + pt $$

2. Find Mean:

Differentiate: $G'_X(t) = p$. Evaluate at $t=1$: $G'_X(1) = p$. Mean $\mu = p$.

3. Find Variance:

Second derivative: $G''_X(t) = 0$. Formula: $Var(X) = 0 + p - p^2 = p(1-p) = pq$. Variance $= pq$.