Paper 2 Section B Practice: Discrete Random Variables

Original practice material

These questions total 21 marks. Show how probability totals and moments are formed.

Answers: Discrete Random Variables Answers.

Question 1: an unknown distribution

A discrete random variable $X$ has the following distribution.

$x$	$0$	$1$	$2$	$3$
$P(X=x)$	$a$	$2a$	$b$	$0.25$

It is given that $E(X)=1.65$.

(a) Form two equations in $a$ and $b$, and hence find $a$ and $b$. [4]

(b) Find $P(0<X\le 2)$. [2]

(c) Find $\mathrm{Var}(X)$. [3]

(d) Write down the value of the cumulative distribution function $F(x)=P(X\le x)$ for $1\le x<2$. [1]

Question 2: transformed rewards and repeated plays

In a game, a player’s score $S$ has distribution

$s$	$-2$	$1$	$4$
$P(S=s)$	$0.3$	$0.5$	$0.2$

(a) Find $E(S)$ and $\mathrm{Var}(S)$. [4]

The monetary reward, in dollars, is $R=3S+5$.

(b) Find $E(R)$ and $\mathrm{Var}(R)$. [2]

The game is played independently three times, with scores $S_1,S_2,S_3$. Let

$$T=S_1+S_2+S_3.$$

(c) Find $E(T)$ and $\mathrm{Var}(T)$. State where independence is used. [3]

(d) Find $P(T=3)$. [2]