Further Math A2s 4.5.2 PGF mean and variance

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The random variable X has the geometric distribution $\operatorname{Geo}(p)$ .

Use the probability generating function of X to show that $\operatorname{Var}(X)=\frac{q}{p^{2}}$ .

Kenny throws an ordinary fair 6-sided dice repeatedly. The random variable X is the number of throws that Kenny takes in order to obtain a 6. The random variable Z denotes the sum of two independent values of X.

$\mathrm{G}^{\prime}(t)=p q t(1-q t)^{-2}+p(1-q t)^{-1}\left[=p(1-q t)^{-2}\right]$
Attempt to differentiate as a product/quotient.

$\mathrm{G}^{\prime \prime}(t)=2 p q^{2} t(1-q t)^{-3}+2 p q(1-q t)^{-2} \quad\left[=2 q p(1-q t)^{-3}\right]$
Attempt to differentiate their $\mathrm{G}^{\prime}(t)$ .

$\mathrm{G}^{\prime}(1)=p(1-q)^{-2}=\frac{1}{p}, \quad \mathrm{G}^{\prime \prime}(1)=\frac{2 q}{P^{2}}$
Substitute t=1 into their $\mathrm{G}^{\prime}(t)$ and $\mathrm{G}^{\prime \prime}(t)$ and use formula
for $\operatorname{Var}(X)$ (at some point).

$\operatorname{Var}(X)=\frac{2 q}{P^{2}}+\frac{1}{p}-\left(\frac{1}{p}\right)^{2}$
(1-q) replaced by p in denominator throughout, and
some cancellation seen (at some point).

$\frac{q}{p^{2}}$
AG
Convincingly obtained.

5

CAIE A-Level Further Math A2 4.5.2 Pgf Mean and Variance Questions

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CAIE A-Level Further Math A2 4.5.2 Pgf Mean and Variance Questions question 1