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
A bag contains a large number of counters. Each counter is numbered 1, 2, 3 or 4

A counter is taken at random from the bag.

The random variable $X$ represents the number on the counter.

The probability distribution of $X$ is given in the following table
$x$ 1 2 3 4
$\mathrm{P}(X = x)$ 0.15 $k$ $2k$ 0.25
where $k$ is a constant.
(a) Find the value of $k$

(2 marks)
(b) Find $\mathrm{P}(X \geqslant 2 \mid X \leqslant 3)$

(2 marks)
Level 2 · 4 marksTry it
2
The discrete random variable $X$ has $P(X = x)$ given by

$\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\\\ \hline P(X=x) & 0.1 & 0.3 & k & 0.2 \end{array}$

Find the value of $k$ .
Level 2 · 2 marksTry it
3
In a game, a player scores 1, 2 or 3 points on each turn. The number of points, $W$ , scored on a randomly chosen turn has cumulative distribution function

$\mathrm{F}(w) = \frac{w(w + 1)}{12} \qquad w = 1, 2, 3$

Find $\mathrm{E}(W)$

(4 marks)
Level 3 · 4 marksTry it
4
SECTION B: STATISTICS

The discrete random variable $X$ has probability distribution

$P(X = x) = kx$ , $\quad x = 1, 2, 3, 4, 5$ ,

where $k$ is a constant.
(a) Show that $k = \dfrac{1}{15}$ . (2 marks)
(b) Find $P(X \ge 4)$ . (2 marks)
(c) Find $P(X \ge 4 \mid X \ge 2)$ . (2 marks)
Level 3 · 6 marksTry it
5
The discrete random variable $X$ has the probability distribution shown in the table, where $k$ is a constant.

$\begin{array}{|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(X = x) & 0.1 & 0.3 & k & 0.2 \\ \hline\end{array}$
(a) Find the value of $k$ . (1 mark)
(b) Find $\mathrm{E}(X)$ . (2 marks)
(c) Find $\mathrm{Var}(X)$ . (3 marks)
(d) Find $\mathrm{P}\big(X > \mathrm{E}(X)\big)$ . (1 mark)
Level 3 · 7 marksTry it

$x$	1	2	3	4
$\mathrm{P}(X = x)$	0.15	$k$	$2k$	0.25

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