Further Mathematics · Pearson Edexcel 9FM0 · Option FS1 · Topic 7

Probability generating functions

Probability generating functions, a compact way to store a distribution that makes means, variances and sums of variables quick to find.

3 sub-topics
Taught in Year 13
40 practice questions
3 animated lessons

Practice questions (40) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

7.1Probability generating functionsY13
7.2Mean and variance from PGFsY13
7.3PGF of a sum of independent variablesY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 40 questions in this topic, free, no account needed.

Probability generating functions

Not needed for Year 12Easy

Probability generating functions · Year 13 · 1 mark

$X$ has probability generating function $G(t)$ . Which expression gives $\mathrm{E}(X)$ ?
(1)

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Differentiate once and put $t = 1$ . $B$ : $G'(1)$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Probability generating functions

Not needed for Year 12Medium

Probability generating functions · Year 13 · 5 marks

The probability generating function of the discrete random variable

X

is

G_X(t) = k(1 + t)^4

.

(a)
Find the value of $k$ .
(2)
(b)
Find $\mathrm{P}(X = 1)$ .
(2)
(c)
Write down the distribution of $X$ , stating its parameters.
(1)

Check your answer before you look at the solution

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Part (a)2 marks

The probabilities sum to 1. $G_X(1) = 1$
$k = \frac{1}{16}$

Part (b)2 marks

Expand with the binomial theorem. $\text{coefficient of } t$
$= \frac{1}{4}$

Part (c)1 mark

$(q + pt)^{n}$ is the PGF of $B(n, p)$ . $X \sim \mathrm{B}\left(4, \frac{1}{2}\right)$

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Probability generating functions

Not needed for Year 12Hard

Mean and variance from a PGF · Year 13 · 5 marks

The discrete random variable

X

has probability generating function

G_X(t)

{}= \dfrac{\frac{1}{2}t^2}{1 - \frac{1}{2}t}

.

(a)
Find $\mathrm{E}(X)$ .
(2)
(b)
Find $\operatorname{Var}(X)$ .
(3)

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Part (a)2 marks

$G'_X(t)$ ${}= \frac{t \left(4 - t\right)}{\left(t - 2\right)^{2}}$
$\mathrm{E}(X) = G'_X(1) = 3$

Part (b)3 marks

Differentiate again. $G''_X(1) = 8$
$\operatorname{Var}(X)$ ${}= G''(1) + G'(1) - [G'(1)]^2$ ${}= 2$

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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