CAIE A-Level Further Math A2 4.5.1 Probability Generating Functions Questions

Practise using probability generating functions to derive distributions, combine independent variables and calculate probabilities with Further Mathematics AS questions and mark schemes.

Exam points

Use probability generating functions to derive distributions and probabilities.

[Maximum number: 3]

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

Show that the probability generating function of X is $\frac{p t}{1-q t}$ , where q=1-p.

$\mathrm{P}(X=r)=p(1-p)^{r-1}=p q^{r-1}$
Implied by $\mathrm{G}(t)=\sum p q^{r-1} t^{r}$ .

$\mathrm{G}(t)=\sum p t(q t)^{r-1}$
(Infinite summation) with common ratio ( q t ) and first
term or common factor pt indicated or GP identified.

$\mathrm{G}(t)=p t\left(\frac{1}{1-q t}\right)=\frac{p t}{1-q t}$
AG
Convincingly obtained.

3

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CAIE A-Level Further Math A2 4.5.1 Probability Generating Functions Questions question 1