What is a discrete random variable in the context of Cambridge International A Levels Mathematics?

In Cambridge International A Levels Mathematics, a discrete random variable is a type of random variable that can take on a finite or countably infinite set of distinct values. These values are typically whole numbers, and each value has a specific probability associated with it. Discrete random variables are fundamental in Probability and Statistics 1, where they are used to model scenarios like the number of heads in coin tosses or the number of students passing an exam.

How do you calculate the expected value of a discrete random variable?

The expected value of a discrete random variable, often denoted as E(X), is calculated by summing the products of each possible value of the variable and its corresponding probability. Mathematically, it is expressed as E(X) = Σ [x * P(x)], where x represents the possible values of the random variable and P(x) is the probability of each value. This concept is crucial in Probability and Statistics 1 for predicting the average outcome of a random process over many trials.

What is the difference between a probability mass function and a cumulative distribution function for discrete random variables?

For discrete random variables, the probability mass function (PMF) gives the probability that the random variable is exactly equal to a specific value. In contrast, the cumulative distribution function (CDF) provides the probability that the random variable is less than or equal to a certain value. Understanding these functions is essential in the Cambridge International A Levels curriculum as they are used to describe the distribution and behavior of discrete random variables in Probability and Statistics 1.

Can you explain the concept of variance for a discrete random variable?

Variance is a measure of how much the values of a discrete random variable deviate from the expected value. It is calculated as the expected value of the squared deviations from the mean, expressed as Var(X) = Σ [(x - E(X))^2 * P(x)], where x represents the possible values, E(X) is the expected value, and P(x) is the probability of each value. In Probability and Statistics 1, variance is used to quantify the spread or dispersion of a random variable's distribution.

How are discrete random variables applied in real-world scenarios?

Discrete random variables are widely used in real-world scenarios to model situations where outcomes are countable and distinct. Examples include the number of defective products in a batch, the number of emails received in an hour, or the number of students scoring above a certain grade. In the Cambridge International A Levels Mathematics syllabus, understanding these applications helps students grasp the practical significance of Probability and Statistics 1 concepts.

Why is "Computing $\text{Var}(X) = E(X^2) - E(X)$" a common mistake in Discrete Random Variables?

Why it happens: Missing squaring. How to avoid it: Var(X) = E(X^2) - [E(X)]^2 — the SECOND term is mean SQUARED. Source: 9709 Examiner Reports 2022-2024

Why is "Using binomial when trials not independent" a common mistake in Discrete Random Variables?

Why it happens: Recognising 'success/failure' but not 'independent'. How to avoid it: Binomial requires: FIXED n, INDEPENDENT trials, SAME p. Sampling WITHOUT replacement violates 'same p'. Source: 9709 Examiner Reports 2022-2024

Probability and Statistics 1Cambridge International A Level Mathematics 9709

Discrete Random Variables

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Short Study Notes — Discrete Random Variables

Start with these resources to cover the key concepts, then work through the practice questions.

Page 1 / 0

Detailed Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Discrete Random Variables, ready to print or save offline.

Discrete Random Variables Study Notes — Cambridge International A Level Mathematics 9709 S1 (2024-2026 syllabus)

Discrete probability distributions. $E(X)$ , $\text{Var}(X)$ . Binomial and geometric distributions.

What you’ll learn

Mapped to the Cambridge International A Level 9709 syllabus (2024-2026).

S1.5.1 — Use discrete probability distributions.
S1.5.2 — Compute $E(X)$ and $\text{Var}(X)$ .
S1.5.3 — Use binomial distribution.
S1.5.4 — Use geometric distribution.

Discrete probability distributions

Table of $x$ and $P(X=x)$ .

Discrete RV $X$ takes values $x_1, x_2, \ldots$ with probabilities $P(X = x_i)$ .

Constraints.

$0 \leq P(X = x_i) \leq 1$ .
$\sum P(X = x_i) = 1$ .

Expectation. $E(X) = \sum x P(X=x)$ . Long-run average.

Variance. $\text{Var}(X) = E(X^2) - [E(X)]^2$ where $E(X^2) = \sum x^2 P(X=x)$ .

Example. Table: $X = 1$ prob $0.2$ , $X=2$ prob $0.5$ , $X=3$ prob $0.3$ .

$E(X) = 0.2 + 1.0 + 0.9 = 2.1$ .
$E(X^2) = 0.2 + 2.0 + 2.7 = 4.9$ .
$\text{Var}(X) = 4.9 - 4.41 = 0.49$ .

Linear transformations. $E(aX + b) = aE(X) + b$ . $\text{Var}(aX + b) = a^2 \text{Var}(X)$ .

Cambridge tip. Always verify probabilities sum to 1 first.

$E(X) = \sum x P(X=x)$ .
$\text{Var}(X) = E(X^2) - [E(X)]^2$ .
Verify $\sum p = 1$ .

See the full worked example for discrete random variables →

Binomial distribution

Counts successes in $n$ trials.

Setup. $n$ INDEPENDENT trials, each success/failure with same probability $p$ . $X$ = number of successes.

Notation. $X \sim B(n, p)$ .

PMF. $P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$ .

Mean and variance. $E(X) = np$ ; $\text{Var}(X) = np(1-p)$ .

Example. $X \sim B(10, 0.3)$ . $P(X = 3) = \binom{10}{3}(0.3)^3(0.7)^7 \approx 0.267$ .

Conditions for binomial.

Fixed $n$ .
Each trial INDEPENDENT.
Each trial has SAME $p$ (so requires replacement or large pool).
Two outcomes per trial (success/failure).

Cambridge tip. State $X \sim B(n, p)$ explicitly.

For X ~ B(10, 0.3) the most likely outcome is r = 3 (the mean np), with probability tailing off in both directions.

$X \sim B(n, p)$ .
PMF: $\binom{n}{r}p^r(1-p)^{n-r}$ .
Mean $np$ , variance $np(1-p)$ .

See the full worked example for discrete random variables →

Geometric distribution

Trials until first success.

Setup. Independent trials with success probability $p$ . $X$ = trial number of first success.

Notation. $X \sim \text{Geo}(p)$ .

PMF. $P(X = r) = (1 - p)^{r-1} p$ , $r = 1, 2, 3, \ldots$

Mean. $E(X) = \frac{1}{p}$ . (Smaller $p$ → longer wait.)

Memoryless property. $P(X > a + b | X > a) = P(X > b)$ . Past failures don't change future.

Example. $X \sim \text{Geo}(0.25)$ .

$P(X = 4) = 0.75^3 \times 0.25 \approx 0.1055$ .
$E(X) = 4$ .

$P(X > r)$ . $= (1-p)^r$ (all $r$ trials fail). Easy formula.

Cambridge tip. State $X \sim \text{Geo}(p)$ explicitly.

$X \sim \text{Geo}(p)$ .
PMF: $(1-p)^{r-1}p$ .
Mean $1/p$ .

See the full worked example for discrete random variables →

How it’s examined

DRVs appear every S1 — typically 12-15 marks. Most-tested: binomial (8-10 marks), expectation table (5-7 marks), geometric (5-7 marks).

Sources: Cambridge International A Level Mathematics 9709 syllabus (2024-2026); 9709 Examiner Reports 2022-2024; 9709/52 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Discrete Random Variables, ready to print or save as PDF.

Step-by-step worked examples — Discrete Random Variables

Step-by-step solutions to past-paper-style questions on discrete random variables, written exactly the way a tutor would explain them at the board.

1 $E(X)$ and $\text{Var}(X)$ from table (8 marks)
Extended• Adapted from 9709/52 May/Jun 2024• expectation
Question
RV $X$ has distribution: $P(X=1) = 0.2$ , $P(X=2) = 0.5$ , $P(X=3) = 0.3$ . Find $E(X)$ and $\text{Var}(X)$ . (8 marks)
Step-by-step solution
1. Step 1
  $E(X) = \sum x P(X=x)$ .
  $E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1 + 0.9 = 2.1$
2. Step 2
  $E(X^2) = \sum x^2 P(X=x)$ .
  $E(X^2) = 1(0.2) + 4(0.5) + 9(0.3) = 0.2 + 2 + 2.7 = 4.9$
3. Step 3
  $\text{Var}(X) = E(X^2) - [E(X)]^2$ .
  $\text{Var}(X) = 4.9 - 4.41 = 0.49$
Answer
$E(X) = 2.1$ ; $\text{Var}(X) = 0.49$ .
2Binomial distribution (6 marks)
Extended• binomial
Question
$X \sim B(10, 0.3)$ . Find $P(X = 3)$ . (6 marks)
Step-by-step solution
1. Step 1
  Binomial PMF. $P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$ .
2. Step 2
  Substitute $n = 10$ , $p = 0.3$ , $r = 3$ .
  $P(X=3) = \binom{10}{3}(0.3)^3(0.7)^7 = 120 \times 0.027 \times 0.0824 \approx 0.267$
Answer
$P(X = 3) \approx 0.267$ .
3Geometric distribution (5 marks)
Extended• geometric
Question
$X \sim \text{Geo}(0.25)$ . Find $P(X = 4)$ and $E(X)$ . (5 marks)
Step-by-step solution
1. Step 1
  Geometric PMF. $P(X = r) = (1-p)^{r-1} p$ , $r = 1, 2, \ldots$
  $P(X = 4) = 0.75^3 \times 0.25 = 0.1055$
2. Step 2
  Expectation. $E(X) = \frac{1}{p}$ .
  $E(X) = \dfrac{1}{0.25} = 4$
Answer
$P(X = 4) \approx 0.1055$ ; $E(X) = 4$ .

Key Formulae — Discrete Random Variables

The formulae you need to memorise for discrete random variables on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.

Expectation
$E(X) = \sum x P(X = x)$
When to use
Mean of a discrete RV.
Variance
$\text{Var}(X) = E(X^2) - [E(X)]^2, \quad E(X^2) = \sum x^2 P(X=x)$
When to use
Spread of discrete RV.
Binomial PMF
$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$
$n$
number of trials
$p$
probability of success
When to use
Number of successes in $n$ independent trials. $X \sim B(n, p)$ .
Binomial mean and variance
$E(X) = np, \quad \text{Var}(X) = np(1-p)$
When to use
For $X \sim B(n, p)$ .
Geometric PMF
$P(X = r) = (1 - p)^{r - 1} p, \quad r = 1, 2, \ldots$
When to use
Trials until first success. $X \sim \text{Geo}(p)$ . $E(X) = 1/p$ .

Key Definitions and Keywords — Discrete Random Variables

Definitions to memorise and the exact keywords mark schemes credit for discrete random variables answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.

Discrete random variable
Examiner keyword
Random variable taking countable distinct values, each with a probability summing to 1.
Expectation (mean)
Examiner keyword
$E(X) = \sum x P(X=x)$ . Long-run average value.
Variance
Examiner keyword
$\text{Var}(X) = E(X^2) - [E(X)]^2$ . Measure of spread of RV.
Binomial distribution
Examiner keyword
Number of successes in $n$ independent trials with success probability $p$ . Notation $X \sim B(n, p)$ .
Geometric distribution
Examiner keyword
Number of trials until first success. $P(X=r) = (1-p)^{r-1}p$ . $E(X) = 1/p$ .

Common Mistakes and Misconceptions — Discrete Random Variables

The traps other students keep falling into on discrete random variables questions — taken from recent Cambridge International A Level 9709 examiner reports and mark schemes — and how to avoid them.

✕Computing $\text{Var}(X) = E(X^2) - E(X)$
9709 Examiner Reports 2022-2024
Why it happens
Missing squaring.
How to avoid it
$\text{Var}(X) = E(X^2) - [E(X)]^2$ — the SECOND term is mean SQUARED.
✕Using binomial when trials not independent
9709 Examiner Reports 2022-2024
Why it happens
Recognising 'success/failure' but not 'independent'.
How to avoid it
Binomial requires: FIXED $n$ , INDEPENDENT trials, SAME $p$ . Sampling WITHOUT replacement violates 'same $p$ '.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Discrete Random Variables — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Discrete Random Variables

Short Study Notes in the form of Slides

Short Study Notes — Discrete Random Variables

Detailed Notes

What you’ll learn

How it’s examined

1E(X)E(X)E(X) and Var(X)\text{Var}(X)Var(X) from table (8 marks)

2Binomial distribution (6 marks)

3Geometric distribution (5 marks)

Expectation

Variance

Binomial PMF

Binomial mean and variance

Geometric PMF

Discrete random variable

Expectation (mean)

Variance

Binomial distribution

Geometric distribution

✕Computing Var(X)=E(X2)−E(X)\text{Var}(X) = E(X^2) - E(X)Var(X)=E(X2)−E(X)

✕Using binomial when trials not independent

Practice questions

Past paper style quiz

Assess your understanding

Discrete Random Variables — frequently asked questions

1 $E(X)$ and $\text{Var}(X)$ from table (8 marks)

✕Computing $\text{Var}(X) = E(X^2) - E(X)$