PGFs of Standard Distributions (Edexcel A-Level Further Mathematics): Flashcards

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

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

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

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

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

$\mathbb{E}[X] = G_X'(1)$

$\text{Var}(X) = G_X''(1) + G_X'(1) - [G_X'(1)]^2$

$\mathbb{E}[X] = np$

$\text{Var}(X) = \lambda$

$|t| < \frac{1}{1-p}$