Discrete Random Variables: Question 5

Syllabus 5.4

Structured AS 5 marks

At a school fair, a tombola stall gives out a token to every player. Each token shows a number of points, and the random variable $X$ represents the number of points on a token drawn at random, with possible values $0$ , $1$ , $3$ and $5$ . The probability distribution of $X$ is shown in the table below, where $k$ is a constant.

$x$	0	1	3	5
$P(X=x)$	$0.4$	$0.3$	$k$	$0.1$

(a) Find the value of $k$ . [2]

(b) Find $E(X)$ . [3]

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Part (a): Finding the unknown probability

All probabilities in the distribution must sum to $1$ : $P(X=0)+P(X=1)+P(X=3)+P(X=5)=1$ $0.4+0.3+k+0.1=1$

Add the known probabilities: $0.4+0.3+0.1=0.8$

So $0.8+k=1 \implies k=1-0.8=0.2$

Check: $0.4+0.3+0.2+0.1=1.00$ ✓, and each value lies between $0$ and $1$ , so this is a valid probability distribution.

Part (b): Finding E(X)

$E(X)=\sum xP(X=x)=0(0.4)+1(0.3)+3(0.2)+5(0.1)$ $=0+0.3+0.6+0.5=1.4$

Check by grouping differently: $1(0.3)+5(0.1)=0.3+0.5=0.8$ , and $3(0.2)=0.6$ , so $E(X)=0.8+0.6=1.4$ , which agrees with the value found above.

Final answers

(a) $k=\boxed{0.2}$
(b) $E(X)=\boxed{1.4}$

Worked solution

Part (a): Finding the unknown probability

Part (b): Finding E(X)

Final answers