Probability generating functions — chapter review — 9231 FPS
Read probabilities from coefficients and recover moments using derivatives. Multiply PGFs only when the variables are independent.
Notes and key methods
For a nonnegative integer-valued X, \(G_X(t)=E(t^X)=\sum p_xt^x\). The coefficient of \(\displaystyle t^{r}\) is \(\displaystyle P(X=r)\), and \(\displaystyle G(1)=1\).
Use \(E(X)=G'(1)\) and \(\operatorname{Var}(X)=G''(1)+G'(1)-[G'(1)]^2\). The second derivative gives E(X(X−1)), not \(\displaystyle E(X^{2})\), which explains the extra first-moment term.
If X and Y are independent nonnegative integer variables, \(G_{X+Y}(t)=G_X(t)G_Y(t)\). For m independent copies of X, use \([G_X(t)]^m\).
Differentiate the product to recover moments, or add independent expectations and variances. A deterministic shift c gives \(G_{X+c}(t)=t^cG_X(t)\); multiplying the value of X by c instead gives \(G_{cX}(t)=G_X(t^c)\).
Check the conditions
X takes nonnegative integer values for an ordinary PGF.
Moments require the relevant derivatives at \(\displaystyle t=1\) to exist.
Independence is essential for PGF multiplication.
A shift and a scale of the variable affect the PGF differently.
Three worked examples
Example 1 · Probability generating functions
X has PGF \(\displaystyle G_{X}(t)= \frac{5+ 6t+ 8t^{2}}{19}\).
\(\displaystyle X_{1}\),…,\(\displaystyle X_{m}\) are independent \(\displaystyle \operatorname{Bin}(4,\frac{1}{5})\) variables, with \(\displaystyle m=4\). Let S be their sum. Thus \(G_S(t)=(1-p+pt)^{16}\).