Probability generating functions
Probability generating functions
- The probability generating function (PGF) of a discrete variable $X$:
$$G(t) = E(t^X) = \sum_x P(X = x)\,t^x$$
- It packs the whole distribution into one function.
- Mean and variance from its derivatives at $t = 1$: $E(X) = G'(1)$ and $\text{Var}(X) = G''(1) + G'(1) - \big(G'(1)\big)^2$.
- The PGF of a sum of independent variables is the product of their PGFs.
Practice
X has G(t) = 0.5 + 0.3t + 0.2t². Since E(X) = G′(1), and G′(t) = 0.3 + 0.4t, what is E(X)?
G′(1) = 0.3 + 0.4(1) = 0.7, so E(X) = 0.7.
Practice
From a PGF, the mean E(X) equals:
E(X) = G′(1), the first derivative evaluated at t = 1.
Practice
The PGF of a sum of independent variables is the product of their PGFs.
Independence makes the generating function of the sum multiply.
You've got it
Key idea
- PGF: $G(t) = E(t^X) = \sum P(X=x)\,t^x$
- $E(X) = G'(1)$; $\text{Var}(X) = G''(1) + G'(1) - (G'(1))^2$
- the PGF of a sum of independent variables is the product of their PGFs