CAIE A-Level Further Math 4.5.2 PGF mean & variance

Practise using derivatives of PGFs to calculate E(X) and Var(X), including algebraic forms with unknown parameters.

Exam points

use G''(1)+G'(1)-[G'(1)]^2 to calculate or prove Var(X)

[Maximum number: 2]

Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by X.

Use the probability generating function of Z to find E(Z).

$\mathrm{G}_{Z}^{\prime}(t)=\frac{1}{144}\left(40+102 t+93 t^{2}+36 t^{3}+5 t^{4}\right)$

M1

Differentiate

$\mathrm{E}(Z)=\mathrm{G}_{Z}^{\prime}(1)=\frac{23}{12}=(=1.92)$

A1

2

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4.5.2—PGF mean and variance question 1