Edexcel A-Level Mathematics AS S1.5 Discrete Random Variables Questions

Practise using probability tables, cumulative functions, E(X), Var(X) and discrete uniform models for small discrete random variables.

Exam points

find missing probabilities or F(x) values from a probability table or cumulative distribution
calculate E(X), E(X^2) and Var(X), including transformations
recognise a discrete uniform distribution and use equal probabilities or its mean in context

Covered syllabus points

S1.5.1 - concept of a discrete random variable S1.5.2 - Probability and cumulative distribution functions S1.5.3 - Mean and variance of a discrete S1.5.4 - discrete uniform distribution

[Maximum number: 7]

The discrete random variable X has probability function p(x) and cumulative distribution function F(x) given in the table below.

\begin{tabular}{|c|c|c|c|c|c|}
\hline x & 1 & 2 & 3 & 4 & 5 \\
\hline p(x) & 0.10 & a & 0.28 & c & 0.24 \\
\hline F(x) & 0.10 & 0.26 & b & 0.76 & d \\
\hline
\end{tabular}

Two independent observations, $X_1$ and $X_2$ , are taken from the distribution of X.

Figure for Question 1 — Edexcel A-Level Mathematics AS

(a)

Write down the value of d

[ 1 ]

(b)

Find the values of a, b and c

[ 3 ]

(c)

Write down the value of P(X>4).

[ 1 ]

(d)

Find the probability that $X_1$ and $X_2$ are both odd.

[ 2 ]

$P(X\text{ is odd})=0.10+0.28+0.24=0.62$ $P(X_1\text{ and }X_2\text{ are both odd})=0.62^2=0.3844$
awrt 0.384

[Maximum number: 11]

The random variable A represents the score when a spinner is spun. The probability distribution for A is given in the following table.

Table for Question 2 — Edexcel A-Level Mathematics AS

(a)

Show that E(A)=3.5

[ 2 ]

$[\mathrm{E}(A)=] 1 \times 0.4+4 \times 0.2+5 \times 0.25+7 \times 0.15$ $=\underline{3.5} \quad(*)$

(b)

Find $\operatorname{Var}(A)$

The random variable B represents the score on a 4 -sided die. The probability distribution for B is given in the following table where k is a positive integer.

Table for Question (b) — Edexcel A-Level Mathematics AS

[ 3 ]

$\left[\mathrm{E}\left(A^{2}\right)=\right] 1 \times 0.4+4^{2} \times 0.2+5^{2} \times 0.25+7^{2} \times 0.15[=17.2]$ $\operatorname{Var}(A)=\mathrm{E}\left(A^{2}\right)-[\mathrm{E}(A)]^{2}=17.2-3.5^{2}$ $=\underline{4.95}$

(3)

(c)

Write down the name of the probability distribution of B.

[ 1 ]

(d)

Given that E(B)=E(A) state, giving a reason, the value of k.

The random variable $X \sim \mathrm{~N}\left(\mu, \sigma^{2}\right)$

Sam and Tim are playing a game with the spinner and the die.

They each spin the spinner once to obtain their value of A and each roll the die once to obtain their value of B.
Their value of A is taken as their value of $\mu$ and their value of B is taken as their value of $\sigma$ . The person with the larger value of P(X>3.5) is the winner.

[ 1 ]

(e)

Find the largest value of P(X>3.5) achievable in this game.

[ 4 ]

Need largest possible $\mu=7$ and smallest possible $\sigma=1$

P(X>3.5) is then $\mathrm{P}\left(Z>\frac{3.5-7}{1}\right)=\mathrm{P}(Z>-3.5)$ $=\underline{0.9998}$ (tables) or 0.999767...(calc)

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)