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
An office receives post every weekday. The number of parcels, $X$ , delivered to the office on a randomly chosen weekday has the cumulative distribution function given in the table.

$\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{F}(x) & 0.05 & k & 2k & 0.85 & k + 0.7 \end{array}$
(a) Find the value of the constant $k$

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

(3 marks)

The number of letters, $Y$ , delivered to the office on a randomly chosen weekday has the probability distribution below, where $a$ and $b$ are constants.

$\begin{array}{c|ccccc} y & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(Y = y) & 0.05 & b & a & 2a & b \end{array}$

The mean number of letters delivered on a weekday is 2.5
(c) By forming a pair of simultaneous equations in $a$ and $b$ , show that $a = 0.25$

(Solutions relying on calculator technology are not acceptable.)

(3 marks)
(d) Show that $\mathrm{E}(Y^2) = 7.2$

(2 marks)
(e) Find $\mathrm{Var}(9 - 2Y)$

(3 marks)

The number of parcels and the number of letters delivered on any weekday are independent.
(f) Find the probability that, on a randomly chosen weekday, the total number of parcels and letters delivered to the office is 7

(2 marks)
Level 5 · 14 marksTry it
2
Mateo has two packs of cards. Pack $A$ contains five cards numbered 1, 2, 3, 4 and 5, and pack $B$ contains four cards numbered 2, 4, 6 and 8.

The discrete random variable $X$ is the number on a card taken at random from pack $A$ . The discrete random variable $Y$ is the number on a card taken at random from pack $B$ .
(a) Write down the name of the probability distribution of $X$

(1 mark)
(b) Find $\mathrm{P}(X \leqslant 3)$

(1 mark)
(c) Write down the value of

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

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

(2 marks)
(d) Showing your working, find $\mathrm{Var}(Y)$

(3 marks)

Mateo takes one card at random from each pack.
(e) Find $\mathrm{P}(X + Y \geqslant 10)$

(2 marks)
(f) Find $\mathrm{P}(X > Y)$

(3 marks)

The random variable $M$ is the larger of the two numbers on Mateo's cards, or that number if both cards show the same number. The cumulative distribution function of $M$ is

$\begin{array}{c|cccccc} m & 2 & 3 & 4 & 5 & 6 & 8 \\ \hline \mathrm{F}(m) & \dfrac{1}{10} & \dfrac{3}{20} & p & \dfrac{1}{2} & \dfrac{3}{4} & 1 \end{array}$
(g) Find the value of $p$ . You must show your working.

(3 marks)
Level 5 · 15 marksTry it
3
The only possible values of the discrete random variable $X$ are $1$ , $2$ , $3$ and $4$ , and for each of these values

$\mathrm{P}(X = x) = k(2x - 1),$

where $k$ is a constant.
(a) Show that $\mathrm{P}(X = 3) = \dfrac{5}{16}$ .

(3 marks)
(b) Show in a table the values of $X$ and their probabilities.

(2 marks)

The random variable $T$ is the sum of two independent observations of $X$ .
(c) (i) Find $\mathrm{P}(T = 6)$ .

(ii) On one occasion, $T = 6$ . Determine the probability that, on this occasion, the two observations of $X$ were equal.

(7 marks)
Level 5 · 12 marksTry it
4
In this question you must show detailed reasoning.

A game uses a spinner with four sectors, numbered $1$ , $2$ , $3$ and $4$ . The number, $X$ , on which the spinner lands is modelled by the probability distribution

$\mathrm{P}(X = x) = \begin{cases} k(x + 2) & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}$

where $k$ is a constant. The results of different spins are independent.
(a) Show that $k = \dfrac{1}{18}$ .

(2 marks)

The spinner is spun three times.
(b) Given that the total of the three numbers obtained is $8$ , determine the probability that the three numbers are all different.

(6 marks)

Harvey spins the spinner repeatedly, stopping as soon as it has landed on $4$ for the second time.
(c) Determine the probability that Harvey's sixth spin is his last.

(3 marks)
Level 5 · 11 marksTry it