Probability Generating Functions (Edexcel A Level Further Maths: Further Statistics 1): Exam Questions

The discrete random variable $X$ has probability generating function

$G_X\left(t\right)=\frac{t^2}{(3-2t{)}^2}$

Specify the distribution of $X$

A fair die is rolled repeatedly.

Describe an outcome that could be modelled by the random variable $X$

Use calculus and $G_X\left(t\right)$ to find

(i) $\text{E}\left(X\right)$

(ii) $\text{Var}\left(X\right)$

The discrete random variable $Y$ has probability generating function

$G_Y\left(t\right)=\frac{t^{10}}{(3-2t^3{)}^2}$

Find the exact value of $\text{P}(Y=19)$

The discrete random variable $V$ has probability distribution

Show that the probability generating function of $V$ is

$G_V\left(t\right)=t^2{\left(\frac{2}{5}t+\frac{3}{5}\right)}^2$

The discrete random variable $W$ has probability generating function

$G_W\left(t\right)=t{\left(\frac{2}{5}t+\frac{3}{5}\right)}^5$

Use calculus to find

(i) $\text{E}\left(W\right)$

(ii) $\text{Var}\left(W\right)$

Given that $V$ and $W$ are independent,

find the probability generating function of $X=V+W$ in its simplest form.

The discrete random variable $Y=2X+3$

Find the probability generating function of $Y$

Find $\text{P}(Y=15)$

The probability generating function of the random variable $X$ is

$G_X\left(t\right)=k(1+2t{)}^5$

where $k$ is a constant.

Show that $k=\frac{1}{243}$

Find $\text{P}(X=2)$

Find the probability generating function of $W=2X+3$

The probability generating function of the random variable $Y$ is

$G_Y\left(t\right)=\frac{t(1+2t{)}^2}{9}$

Given that $X$ and $Y$ are independent,

find the probability generating function of $U=X+Y$ in its simplest form.

Use calculus to find the value of $\text{Var}\left(U\right)$

A discrete random variable $X$ has probability generating function given by

$G_X\left(t\right)=\frac{1}{64}(a+b{t}^2{)}^2$

where $a$ and $b$ are positive constants.

Write down the value of $\text{P}(X=3)$

Given that $\text{P}(X=4)=\frac{25}{64}$

(i) Find $\text{P}(X=2)$

(7)

(ii) Find $\text{E}\left(X\right)$

(3)

The random variable $Y=3X+2$

Find the probability generating function of $Y$

The discrete random variable $X$ has probability generating function

$G_X\left(t\right)=k \text{ln}\left(\frac{2}{2-t}\right)$

where $k$ is a constant.

Find the exact value of $k$

Find the exact value of $\text{Var}\left(X\right)$

Find $\text{P}(X=3)$

The probability generating function of the discrete random variable $X$ is given by

$G_X\left(t\right)=k(3+t+2t^2{)}^2$

Show that $k=\frac{1}{36}$

Find $\text{P}(X=3)$

Show that $\text{Var}\left(X\right)=\frac{29}{18}$

Find the probability generating function of $2X+1$

The random variable $X$ has probability generating function ${\text{G}}_X\left(t\right)$, where

${\text{G}}_X\left(t\right)=(9-8t{)}^{-\frac{1}{2}}$

Use calculus to find $\text{Var}\left(X\right)$. Show your working clearly.

Find the exact value of $\text{P}(X\le 2)$.

The independent random variables $X_1$, $X_2$, $X_3$ and $X_4$ each have the same distribution as $X$. The random variable $Y$ is defined by

$Y=X_1+X_2+X_3+X_4+2$

By finding the probability generating function of $Y$, state the name of the distribution of $Y$.

Hence, or otherwise, find $\text{P}(X_1+X_2+X_3+X_4=2)$.