What is the difference between a discrete and a continuous random variable?

A discrete random variable can only take specific, separate values (like integers 1, 2, 3...). You can count them. A continuous random variable can take any value within a given range (like a person's height, which could be 1.75m, 1.751m, 1.7512m...).

Why is the expectation E(X) sometimes a value that X can't actually take?

Expectation is the long-run average, not a guaranteed outcome. For a fair die, E(X) = 3.5. This means if you roll it thousands of times, the average of all your scores will be very close to 3.5. It's a theoretical central point of the distribution.

Is there an easy way to remember the variance formula?

Think of it as 'Mean of the squares minus the square of the mean'. In symbols, that's $E(X^2) - [E(X)]^2$. This verbal cue helps many students remember the order and the squaring.

Can a probability in the distribution table be negative or greater than 1?

Absolutely not. A core rule of probability is that for any event A, $0 \le P(A) \le 1$. If you calculate a probability and it's outside this range, you have made a mistake. Similarly, the sum of all probabilities in your distribution must be exactly 1.

Where can I learn Discrete random variables for Cambridge 9709?

This free lesson on MarkScheme covers syllabus point 5.4 (Discrete random variables) for Cambridge A-Level Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9709/5-4-discrete-random-variables.

Is Discrete random variables (5.4) on 9709 free to study?

Yes. The Mathematics 9709 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Discrete random variables.

Discrete random variables 9709 — Free…

Discrete random variables

In probability, we often deal with outcomes that are not just 'win' or 'lose', but have numerical values. For example, the score on a spinner, the number of defective items in a batch, or the number of goals in a football match. This lesson explores how to mathematically model these situations using discrete random variables, allowing us to predict average outcomes and understand the variability involved.

BY THE END, YOU CAN…

1Understand and define a discrete random variable and its probability distribution.
2Calculate and interpret the expectation E(X) of a discrete random variable.
3Calculate the variance Var(X) and standard deviation of a discrete random variable.
4Apply the rules for expectation and variance to a linear transformation of a random variable, aX + b.

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Blueprint of Chance

A discrete random variable is just a number that changes by chance, like the score on a die. We can map out its probabilities and calculate its average outcome (expectation) and how spread out the results are (variance).

THINK OF IT LIKE…

Imagine a game where you're paid based on a die roll. A discrete random variable is the 'score' you get. The probability distribution is the 'rulebook' telling you how likely each score is. The expectation is the average amount of money you'd expect to win per game if you played thousands of times.

1
A discrete variable X has specific values 'x'. Each value has a probability P(X=x), and all probabilities must sum to 1.
2
The expectation E(X) is the weighted average. Multiply each value 'x' by its probability and sum them up: E(X) = Σx P(X=x).
3
The variance Var(X) measures spread. First find E(X²), the average of the squared values, then use Var(X) = E(X²) − [E(X)]².
4
If you transform X into a new variable Y = aX + b, the new expectation is E(Y) = aE(X) + b, but the new variance is Var(Y) = a²Var(X).

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$E(X) = \mu = \sum x P(X=x) \\ \\ Var(X) = \sigma^2 = E[(X - \mu)^2] = \sum (x-\mu)^2 P(X=x) \\ \\ \text{A more useful formula for calculation is:} \\ Var(X) = E(X^2) - [E(X)]^2 \\ \text{where } E(X^2) = \sum x^2 P(X=x)$

Tap a symbol — great for exam definitions

$E(aX + b) = aE(X) + b \\ \\ Var(aX + b) = a^2 Var(X)$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Defining a Discrete Random Variable

A random variable is a variable whose value is a numerical outcome of a random phenomenon. We use a capital letter, like $X$ , to denote a random variable. A discrete random variable is one that can only take on a finite or countably infinite number of distinct values. For example, the score when rolling a standard six-sided die can only be {1, 2, 3, 4, 5, 6}. You cannot roll a 2.5. In contrast, a continuous random variable, which we will see later, can take any value in a given range (e.g., height or weight).

Probability Distributions

A probability distribution for a discrete random variable is a list, table, or formula that specifies all possible values of the variable and their corresponding probabilities. It's a complete picture of the random variable's behaviour. We denote the probability that the random variable $X$ takes a specific value $x$ as $P(X=x)$ .

For any valid probability distribution, two conditions must be met:
1. The probability of each outcome must be between 0 and 1 inclusive: $0 \le P(X=x) \le 1$ .
1. The sum of all probabilities must be exactly 1: $\sum P(X=x) = 1$ .

Expectation and Variance

The expectation of a random variable, denoted $E(X)$ or $\mu$ , is its theoretical mean value. It's what you'd expect the average outcome to be if you repeated the random experiment an infinite number of times. The variance, denoted $Var(X)$ or $\sigma^2$ , measures the spread of the distribution. It tells us, on average, how far the values are from the mean. A small variance means the outcomes are clustered tightly around the expectation, while a large variance means they are more spread out.

EXAM TIP

Always show your working for $E(X)$ and $Var(X)$ . While you can use your calculator's statistics mode to verify your final answer, you won't get method marks in an exam without showing the ' $\sum x p$ ' and ' $\sum x^2 p$ ' calculations.

Linear Transformations: E(aX + b) and Var(aX + b)

Often in real-world problems, we need to adjust a random variable. For instance, if $X$ is a score in a game, $2X$ might be the prize money in pounds. If there is a £5 entry fee, the profit would be $2X - 5$ . This is a linear transformation of the form $aX+b$ . Fortunately, there are simple rules to find the new expectation and variance without having to recalculate from the probability distribution.

$E(aX + b) = aE(X) + b \\ \\ Var(aX + b) = a^2 Var(X)$

Expectation is linear. Both scaling by 'a' and shifting by 'b' affect the mean in a straightforward way.
Variance is only affected by scaling. Shifting the data by 'b' moves the entire distribution but doesn't change its spread, so 'b' has no effect on variance.
The scaling factor 'a' is squared for variance ( $a^2$ ) because variance is calculated using squared differences. The standard deviation is therefore scaled by $|a|$ : $SD(aX+b) = |a|SD(X)$ .

EXAM TIP

A very common error is forgetting to square the 'a' term when calculating $Var(aX+b)$ . Always write $a^2Var(X)$ to remind yourself. Another trap is when 'a' is negative, for example $Var(3-2X)$ . Here $a=-2$ , so $a^2 = (-2)^2 = 4$ . The variance will be $4Var(X)$ .

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

A biased four-sided die has scores 1, 2, 3, and 4. The probability distribution of the score, $X$ , is given in the table. The probability of scoring a 2 is twice the probability of scoring a 1.

x	1	2	3	4
P(X=x)	p	2p	0.4	0.1

(i) Find the value of $p$ . (ii) Find $E(X)$ .

1
(i) The sum of probabilities must be 1. $\sum P(X=x) = 1$ $p + 2p + 0.4 + 0.1 = 1$ $3p + 0.5 = 1$ $3p = 0.5$ $p = \frac{0.5}{3} = \frac{1}{6}$

EXAMPLE 2

A discrete random variable $Y$ has the following probability distribution:

y	0	1	2	3
P(Y=y)	0.1	0.3	0.4	0.2

(i) Find $E(Y)$ and $Var(Y)$ . (ii) A new random variable $Z$ is defined as $Z = 5Y - 2$ . Find $E(Z)$ and $Var(Z)$ .

1
(i) First, calculate $E(Y)$ . $E(Y) = \sum y P(Y=y)$ $E(Y) = (0 \times 0.1) + (1 \times 0.3) + (2 \times 0.4) + (3 \times 0.2)$ $E(Y) = 0 + 0.3 + 0.8 + 0.6 = 1.7$

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEADiscrete random variables

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 11

What is a discrete random variable?

TAP OR SPACE TO FLIP

ANSWER

A variable whose value is a numerical outcome of a random phenomenon, which can only take a countable number of values.

TAP OR SPACE TO FLIP BACK

1 / 11 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
For any valid probability distribution, two conditions must be met:
✓
1. The probability of each outcome must be between 0 and 1 inclusive: $0 \le P(X=x) \le 1$ .
✓
1. The sum of all probabilities must be exactly 1: $\sum P(X=x) = 1$ .

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS5.3 Probability NEXT →5.5 The normal distribution

Discrete random variables

BY THE END, YOU CAN…

1Understand and define a discrete random variable and its probability distribution.
2Calculate and interpret the expectation E(X) of a discrete random variable.
3Calculate the variance Var(X) and standard deviation of a discrete random variable.
4Apply the rules for expectation and variance to a linear transformation of a random variable, aX + b.

P(X=x)

0.4

0.1

P(Y=y)

0.1

0.3

0.4

0.2

Revise Discrete random variables for Cambridge 9709 (5.4)

In simple terms

Explore the concept

Key formulas

Full topic notes

Defining a Discrete Random Variable

Probability Distributions

Expectation and Variance

Linear Transformations: E(aX + b) and Var(aX + b)

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Defining a Discrete Random Variable

Probability Distributions

Expectation and Variance

Linear Transformations: E(aX + b) and Var(aX + b)

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Discrete random variables