A-level · Statistical distributions

Discrete random variables and probability distributionsExam-style questions

What a discrete random variable is, and how a probability distribution lists each value it can take with the probability of that value — as a table or as a probability function. The key rule that the probabilities must sum to 1 (used to find an unknown), reading cumulative probabilities like P(X ≤ 2) off a distribution, and the special case of a discrete uniform distribution.

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27 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In a computer game, the computer selects one of the numbers 1, 2, 3, 5 and 8. The random variable $X$ represents the number selected, and the probability distribution of $X$ is given in the table below.

$\begin{array}{c|ccccc} x & 1 & 2 & 3 & 5 & 8 \\ \hline \mathrm{P}(X = x) & 0.1 & 0.25 & 0.3 & 0.2 & 0.15 \end{array}$
(a) Find $\mathrm{P}(5 < 3X - 1 \leqslant 14)$

(1 mark)

Given that $\mathrm{E}(X) = 3.7$
(b) show that $\mathrm{Var}(X) = 4.71$

(3 marks)

In one version of the game, a player is awarded $Y$ points for the number selected, where $Y = aX - b$ and $a$ and $b$ are positive constants. The mean of $Y$ is $7.1$ and the variance of $Y$ is $42.39$
(c) Find the value of $a$ and the value of $b$

(4 marks)

Amir and Beth play a different version of the game. The computer selects one number for Amir and, independently, one number for Beth.

Amir's score is the number selected for him.

Beth's score is found by multiplying the number selected for her by 3 and then subtracting 5
(d) Find the probability that Amir's score is greater than Beth's score.

(4 marks)
Level 4 · 12 marksTry it
2
A warehouse receives one delivery each morning. The number of staff, $X$ , available to unload the delivery on a randomly chosen morning is modelled by a discrete random variable with the following probability distribution.

$\begin{array}{c|cccc} x & 2 & 3 & 4 & 6 \\ \hline \mathrm{P}(X = x) & 0.05 & 0.15 & 0.5 & 0.3 \end{array}$
(a) Show that $\mathrm{E}\left(\dfrac{1}{X}\right) = 0.25$

(1 mark)
(b) Find $\mathrm{Var}\left(\dfrac{1}{X}\right)$

(3 marks)

The time, $Y$ hours, taken to unload the delivery is modelled by $Y = \dfrac{12}{X}$
(c) Find

(i) $\mathrm{E}(Y)$

(ii) $\mathrm{Var}(Y)$

(3 marks)

Given that, on a randomly chosen morning, the delivery takes more than $2.5$ hours to unload,
(d) find the probability that at least $3$ staff were available that morning.

(5 marks)
Level 4 · 12 marksTry it
3
A biased spinner has five sectors, labelled 1, 2, 3, 4 and 5.

When the spinner is spun, $X$ is the label of the sector on which it stops. For this spinner, the values of $\mathrm{F}(x)$ , the cumulative distribution function of $X$ , are as follows.

$\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline \mathrm{F}(x) & \dfrac{k}{40} & \dfrac{k + 4}{40} & \dfrac{3k}{40} & \dfrac{4k - 2}{40} & \dfrac{k + 30}{40} \end{array}$

where $k$ is a positive constant.
(a) Show that $k = 10$

(1 mark)
(b) Find the probability distribution of $X$

(3 marks)
(c) Hence write down the mode of $X$

(1 mark)
(d) Calculate $\mathrm{Var}(3X - 5)$

(5 marks)
Level 4 · 10 marksTry it
4
Priya has two fair spinners.
- Spinner $A$ has five sectors numbered 1, 2, 3, 4 and 5
- Spinner $B$ has three sectors numbered 0, 1 and 2
The random variable $X$ $X$ represents the number that spinner $A$ $A$ lands on when it is spun once. The random variable $Y$ $Y$ represents the number that spinner $B$ $B$ lands on when it is spun once.
(a) Write down the name of the distribution of $X$ $X$

(1 mark)
(b) Showing your working, find $\mathrm{Var}(X)$ $Var (X)$

(3 marks)

Priya spins each spinner once.
(c) Find $\mathrm{P}(X + Y \leqslant 3)$ $P (X + Y ⩽ 3)$

(2 marks)

The random variable $T$ $T$ is the product of the number that spinner $A$ $A$ lands on and the number that spinner $B$ $B$ lands on.
(d) Find the probability distribution of $T$ $T$

(4 marks)
Level 4 · 10 marksTry it
5
Six tiles are numbered $1$ , $2$ , $3$ , $4$ , $5$ and $6$ . Three of the tiles are chosen at random, without replacement. The random variable $X$ is the largest of the three numbers on the chosen tiles.
(a) Show that $\mathrm{P}(X = 5) = \dfrac{3}{10}$ .

(2 marks)
(b) Draw up the probability distribution table for $X$ .

(3 marks)
(c) Given that $\mathrm{E}(X) = 5.25$ , find $\mathrm{Var}(X)$ .

(2 marks)
Level 4 · 7 marksTry it

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