# Discrete random variables

> CIE A-Level Mathematics · Probability & Statistics 1 (S1)
> Source: https://www.owlsprep.com/study/cie-9709-u4-discrete-random-variables/

This module covers core properties of discrete random variables, their probability distributions, and the calculation of expectation and variance, a foundational topic for CIE A-Level Statistics 1.

**Prerequisites:** [Basic probability concepts and rules](https://www.owlsprep.com/study/cie-9709-u4-probability/)

## Learning objectives

- Distinguish between discrete and continuous random variables
- Construct and use probability distributions for discrete random variables
- Calculate expected value E(X) and variance Var(X) for discrete random variables

## How often is this tested?

Based on OwlsPrep's analysis of official CIE 9709 S1 past papers (2016–2025): **Discrete random variables** appears **67 times** in the last 10 years — **14.8%** of all S1 questions (67 of 438).

Most-tested forms: Probability distribution table (57), Variance Var(X) (38), Expectation E(X) (25).

## Definition of Discrete Random Variables

A random variable is a quantity whose value depends on the outcome of a random, uncertain event. Discrete random variables take only distinct, separate values (most commonly whole numbers), unlike continuous random variables which can take any value in an interval.

**Discrete Random Variable** — A random variable that can only take a countable number of distinct values, where each possible value has a non-zero probability of occurring.

*Notation:* X

*Example:* Number of heads in 5 coin tosses, number of defective items in a batch, score on a die roll

**Worked example:** Identify which of the following are discrete random variables: (a) Number of customers arriving at a shop in 1 hour, (b) Weight of a randomly selected apple, (c) Number of correct answers on a 10-question multiple choice test.

1. Check if the variable takes countable distinct values:
2. (a) Number of customers can only be 0, 1, 2, ... (whole numbers), so it is countable. Conclusion: discrete.
3. (b) Weight can take any value in a range of possible weights, it is uncountable. Conclusion: not discrete.
4. (c) Number of correct answers is 0 to 10, distinct whole numbers, so countable. Conclusion: discrete.
5. Final answer: (a) and (c) are discrete random variables.

## Probability Distributions

The probability distribution of a discrete random variable \(X\) specifies the probability \(P(X=x)\) for every possible value \(x\) that \(X\) can take. All valid probability distributions follow two core rules.

> **info**
>
> For all valid discrete probability distributions: 1. \(0 \leq P(X=x) \leq 1\) for all \(x\), 2. The sum of all probabilities equals 1: \(\sum P(X=x) = 1\)

**Worked example:** A discrete random variable \(X\) has the probability distribution below. Find the value of the unknown constant \(k\):
| x    | 1 | 2 | 3 | 4 |
|------|---|---|---|---|
| P(X=x) | k | 2k | 0.3 | 0.1 |

1. Use the rule that the sum of all probabilities equals 1, to set up an equation:
2. $$k + 2k + 0.3 + 0.1 = 1$$
3. Simplify and solve for k:
4. $$3k + 0.4 = 1 \rightarrow 3k = 0.6 \rightarrow k = 0.2$$
5. Check all probabilities are between 0 and 1: 0.2, 0.4, 0.3, 0.1, all satisfy the condition. Final answer: \(k=0.2\)

## Expectation and Variance

The expected value (or population mean) of \(X\) is the long-run average value we would expect to get if we repeated the random experiment many times. Variance measures how spread out the values of \(X\) are around the expected value.

**Expected Value** — For a discrete random variable \(X\), the expected value is calculated as: \(E(X) = \sum x P(X=x)\), where the sum is over all possible values of \(x\).

*Notation:* E(X) or \(\mu\)

**Variance** — Variance is calculated using the identity: \(Var(X) = E(X^2) - [E(X)]^2\), where \(E(X^2) = \sum x^2 P(X=x)\).

*Notation:* Var(X) or \(\sigma^2\)

**Worked example:** Using the probability distribution from the previous example (x: 1,2,3,4; P(X=x): 0.2, 0.4, 0.3, 0.1), calculate \(E(X)\) and \(Var(X)\).

1. First calculate \(E(X)\) by multiplying each \(x\) by \(P(X=x)\) and summing:
2. $$E(X) = (1 \times 0.2) + (2 \times 0.4) + (3 \times 0.3) + (4 \times 0.1) = 0.2 + 0.8 + 0.9 + 0.4 = 2.3$$
3. Next calculate \(E(X^2)\) by multiplying each \(x^2\) by \(P(X=x)\) and summing:
4. $$E(X^2) = (1^2 \times 0.2) + (2^2 \times 0.4) + (3^2 \times 0.3) + (4^2 \times 0.1) = 0.2 + 1.6 + 2.7 + 1.6 = 6.1$$
5. Use the variance formula to get the final result:
6. $$Var(X) = E(X^2) - [E(X)]^2 = 6.1 - (2.3)^2 = 6.1 - 5.29 = 0.81$$

## Common pitfalls

- **Wrong:** Writing variance as \(Var(X) = [E(X)]^2 - E(X^2)\)
  - Why it fails: Simple sign error that almost always results in a negative variance, which is impossible
  - Correct: Remember variance is always non-negative, so use: \(Var(X) = E(X^2) - [E(X)]^2\)
- **Wrong:** Calculating \(E(X)\) as (sum of x values) / (number of values), ignoring unequal probabilities
  - Why it fails: Confuses uniform distributions (where all probabilities are equal) with general discrete distributions
  - Correct: Always multiply each \(x\) by \(P(X=x)\) before summing, regardless of whether probabilities are equal or not
- **Wrong:** Calling a rounded continuous variable (e.g. height to nearest cm) discrete
  - Why it fails: Discreteness depends on the underlying variable's possible values, not how it is measured
  - Correct: If the underlying variable can take any value in a range, it is continuous, even when rounded to whole units
- **Wrong:** Forgetting to check that all probabilities are between 0 and 1 after solving for an unknown constant \(k\)
  - Why it fails: Sometimes multiple solutions for \(k\) exist, and one may give an invalid negative probability or probability greater than 1
  - Correct: Always verify that the value of \(k\) you found results in all probabilities between 0 and 1

## Cheatsheet

| Concept | Formula/Rule |
| --- | --- |
| Discrete Random Variable | Countable distinct values, each with non-zero probability |
| Probability Rule | $\sum P(X=x) = 1$, $0 \leq P(X=x) \leq 1$ |
| Expected Value | $E(X) = \sum x P(X=x)$ |
| Expected Value of $X^2$ | $E(X^2) = \sum x^2 P(X=x)$ |
| Variance | $Var(X) = E(X^2) - [E(X)]^2$ |

## What's next

Discrete random variables form the foundation for all specific probability distributions tested in CIE A-Level Statistics. The general expectation and variance rules you learned here apply directly to named distributions such as the binomial, helping you derive and remember their standard formulas. This topic also builds the core concepts you need for continuous random variables: your next steps are the two named discrete distributions — the binomial and geometric — and then the normal distribution, the most important continuous distribution in the course. These same ideas underpin topics like sampling and hypothesis testing further into your A-Level studies, so mastery of the rules in this module will make all future statistics topics much easier to learn.

- [The Binomial Distribution](https://www.owlsprep.com/study/cie-9709-u4-the-binomial-distribution/)
- [The Geometric Distribution](https://www.owlsprep.com/study/cie-9709-u4-the-geometric-distribution/)
- [Normal distribution](https://www.owlsprep.com/study/cie-9709-u4-normal-distribution/)

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