FS1-7 Probability generating functions — 9FM0 spec coverage

FS1-7.1

Definitions, derivations and applications. Use of the probability generating function for the negative binomial, geometric, binomial and Poisson distributions.

For a non-negative integer-valued random variable, the probability generating function is $G_X(t)=E(t^X)=\sum_xP(X=x)t^x$ .
The standard PGFs are $(1-p+pt)^n$ for binomial, $e^{\lambda(t-1)}$ for Poisson, $\dfrac{pt}{1-(1-p)t}$ for geometric, and $\left(\dfrac{pt}{1-(1-p)t}\right)^r$ for negative binomial.
The coefficient of $t^x$ in $G_X(t)$ equals $P(X=x)$ , so a PGF encodes the whole distribution.
A common error is to omit the factor $t$ in the geometric PGF; the Edexcel convention starts at trial $1$ , not at zero failures.

Tier 1 · Easy

3 marks

ORIGINAL

Given $X\sim\operatorname{Bin}(4,0.3)$ , write down $G_X(t)$ and use its coefficient of $t^2$ to find $P(X=2)$ .

Tier 2 · Standard

5 marks

ORIGINAL

A geometric random variable $X$ has parameter $p=0.4$ and counts the trial of the first success. Derive its PGF and hence find $P(X=5)$ .

Tier 3 · Hard

7 marks

ORIGINAL

The random variable $X$ counts the trial on which the third success occurs in independent trials with success probability $0.25$ . Derive $G_X(t)$ and use it to find $P(X=6)$ .

FS1-7.2

Use to find the mean and variance.

For a PGF $G_X$ , $G'_X(1)=E(X)$ and $G''_X(1)=E[X(X-1)]$ .
Therefore $\operatorname{Var}(X)=G''_X(1)+G'_X(1)-[G'_X(1)]^2$ .
The check $G_X(1)=1$ confirms that the encoded probabilities sum to one before derivatives are used.
A common error is to treat $G''_X(1)$ as $E(X^2)$ ; the missing $G'_X(1)$ must be added.

Tier 1 · Easy

4 marks

ORIGINAL

The PGF of $X$ is $G_X(t)=(0.8+0.2t)^6$ . Use derivatives of the PGF to find $E(X)$ and $\operatorname{Var}(X)$ .

Tier 2 · Standard

5 marks

ORIGINAL

A random variable has PGF $G_X(t)=\dfrac{2t}{3-t}$ . Use the PGF to find its mean and variance.

Tier 3 · Hard

6 marks

ORIGINAL

A random variable has PGF $G_X(t)=k(1+2t+3t^2+4t^3)$ . Find $k$ , then use derivatives to calculate $E(X)$ and $\operatorname{Var}(X)$ .

FS1-7.3

Probability generating function of the sum of independent random variables.

If $X$ and $Y$ are independent, then $G_{X+Y}(t)=G_X(t)G_Y(t)$ .
Multiply and simplify the PGFs before identifying a familiar distribution or extracting coefficients.
For example, multiplying $e^{\lambda(t-1)}$ and $e^{\mu(t-1)}$ gives $e^{(\lambda+\mu)(t-1)}$ , the PGF of $\operatorname{Po}(\lambda+\mu)$ .
A common error is to multiply PGFs without establishing independence; dependence prevents the expectation from factorising.

Tier 1 · Easy

4 marks

ORIGINAL

Independent variables have distributions $X\sim\operatorname{Po}(1.4)$ and $Y\sim\operatorname{Po}(2.6)$ . Use PGFs to identify the distribution of $X+Y$ and find $P(X+Y=0)$ .

Tier 2 · Standard

5 marks

ORIGINAL

Independent variables satisfy $X\sim\operatorname{Bin}(5,0.4)$ and $Y\sim\operatorname{Bin}(7,0.4)$ . Use PGFs to identify $S=X+Y$ and calculate $P(S=3)$ .

Tier 3 · Hard

7 marks

ORIGINAL

Independent geometric variables $X$ and $Y$ each have parameter $0.4$ and count trials to first success. Use PGFs to identify the distribution of $S=X+Y$ and find $P(S>5)$ .