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Probability generating functions
A Probability Generating Function (PGF) is a special function that encodes all the probabilities of a discrete random variable into a single polynomial. By manipulating this function, we can easily extract key information like probabilities, the mean, and the variance without needing the full probability distribution.
Need to know
What you need to know
- The PGF is defined for discrete random variables taking values in ${0, 1, 2, \dots}$.
- The coefficient of $t^r$ in the expansion of $G_X(t)$ is precisely $P(X=r)$.
- Substituting $t=1$ into the PGF gives $G_X(1) = \sum P(X=x) = 1$, which is a useful check.
Explanation
The Probability Vending Machine
- Define the PGF for a discrete random variable $X$ as $G_X(t) = E(t^X) = \sum P(X=x)t^x$.
- Extract probabilities by finding the coefficient of $t^r$ in the expansion of $G_X(t)$, or by using the formula $P(X=r) = \frac{G_X^{(r)}(0)}{r!}$.
- Calculate the mean and variance using derivatives evaluated at $t=1$: $E(X) = G_X'(1)$ and $Var(X) = G_X''(1) + G_X'(1) - [G_X'(1)]^2$.
- For a sum of independent variables, $Z = X+Y$, find the PGF of the sum by multiplying the individual PGFs: $G_Z(t) = G_X(t)G_Y(t)$.