Probability Generating Functions (PGFs) (Edexcel A-Level Further Mathematics): Flashcards

$G_X(t) = \sum_{x=0}^\infty p_x t^x$

A dummy variable (not a probability)

$1$

$\frac{1}{x!} \frac{d^x}{dt^x} G_X(t) \bigg|_{t=0}$

$G_X(t) = [pt + (1-p)]^n$

$G_X(t) = e^{\lambda(t-1)}$

$G_X(t) = \frac{p}{1-(1-p)t}$

$G_X(t) = \left(\frac{p}{1-(1-p)t}\right)^r$

Mean and variance

Differentiate and evaluate at $t=0$