# Discrete Random Variables

> Edexcel International A-Level Mathematics · IAL S1
> Source: https://www.owlsprep.com/study/edexcel-ial-math-s1-discrete-random-variables/

This guide covers all Edexcel IAL S1 content for discrete random variables, including probability functions, CDF calculations, mean/variance with linear transformations, and the discrete uniform distribution, aligned to 2018 Issue 3 syllabus requirements.

**Prerequisites:** [Basic probability rules for discrete events](https://www.owlsprep.com/study/edexcel-ial-math-s1-basic-probability/); [Summation notation for discrete values](https://www.owlsprep.com/study/edexcel-ial-math-s1-summation-basics/)

## Learning objectives

- Define discrete random variables and their probability functions
- Calculate cumulative distribution functions (CDF) from probability distributions
- Compute expected value (E(X)), variance (Var(X)) and apply linear transformation rules
- Use discrete uniform distribution properties to solve S1 exam problems

## What is a Discrete Random Variable?

**Discrete Random Variable (DRV)** — A variable that takes a countable set of distinct numerical values, each associated with a probability of occurrence, from the outcome of a random experiment.

*Example:* The score rolled on a 6-sided die, number of heads in 2 coin flips

Discrete variables have clear gaps between possible values (e.g. you cannot roll a 2.5 on a standard die). Continuous random variables are out of scope for S1 and covered in S2. Non-numerical discrete outcomes (e.g. card suits, colours) must be assigned numerical values to be modelled as DRVs.

**Worked example:** State which of the following are discrete random variables: (a) Height of a randomly selected student, (b) Number of cars passing a checkpoint in 10 minutes, (c) Suit of a randomly drawn playing card.

1. Check if the variable is numerical and countable. (a) Height is a continuous measurement with no distinct countable values → not a DRV.
2. (b) Number of cars takes distinct integer values 0, 1, 2, ... → countable and numerical, so it is a DRV.
3. (c) Suit is non-numerical (hearts, diamonds, etc.) → not a numerical variable, so not a DRV without value assignment.

> **note**
>
> Always assign numerical values to non-numerical outcomes before performing DRV calculations like mean or variance.

*Calculator:* allowed

## Probability Function and Cumulative Distribution Function

**Probability Function (p(x))** — Gives the probability that the discrete random variable X takes the specific value x. All p(x) values satisfy 0 ≤ p(x) ≤ 1 and the sum of all p(x) over all possible x equals 1.

*Notation:* p(x) = P(X = x)

The cumulative distribution function (CDF) calculates the total probability of all outcomes up to a given value. It is a non-decreasing function that ranges from 0 (for values below the minimum possible X) to 1 (for values above the maximum possible X).

**Cumulative Distribution Function (F(x))** — Sum of all probability values for outcomes of X that are less than or equal to x₀.

*Notation:* F(x₀) = P(X ≤ x₀) = \sum_{x ≤ x₀} p(x)

**Worked example:** A discrete random variable X has the probability function p(x) = kx for x = 1, 2, 3. (a) Find the value of k. (b) Find F(2), the CDF at x=2.

1. Use the rule that the sum of all p(x) values equals 1:
2. $$\sum_{x=1}^3 p(x) = k + 2k + 3k = 6k = 1$$
3. Solve for k: k = 1/6
4. Calculate F(2) as the sum of p(1) and p(2):
5. $$F(2) = p(1) + p(2) = 1/6 + 2/6 = 3/6 = 1/2$$

> **Exam tip**
>
> Always verify that the sum of all p(x) values equals 1 to catch arithmetic errors in probability function questions.

*Calculator:* allowed

## Expected Value and Variance of DRVs

**Expected Value (E(X)) and Variance (Var(X))** — E(X) is the long-run mean of the DRV. Var(X) measures the spread of the distribution around the mean, and is always non-negative.

*Notation:* E(X) = \sum x p(x), \quad Var(X) = E(X^2) - [E(X)]^2 = \sum x^2 p(x) - \mu^2

For linear transformations of the form Y = aX + b where a and b are constants, two key rules apply: the mean scales by a and shifts by b, while variance only scales by the square of a (the constant b does not affect spread).

- E(aX + b) = aE(X) + b
- Var(aX + b) = a² Var(X)

**Worked example:** A DRV X has E(X) = 4 and Var(X) = 3. A new variable Y is defined as Y = 2X - 5. Calculate (a) E(Y), (b) Var(Y).

1. Apply the linear expectation rule:
2. $$E(Y) = E(2X - 5) = 2E(X) - 5 = 2(4) - 5 = 3$$
3. Apply the linear variance rule, ignoring the constant -5:
4. $$Var(Y) = Var(2X - 5) = 2^2 Var(X) = 4(3) = 12$$

> **warning**
>
> Never forget to square the coefficient of X when calculating variance of a linear transformation. This is one of the most common marks lost on this topic.

*Calculator:* allowed

## Discrete Uniform Distribution

**Discrete Uniform Distribution** — A DRV X follows a discrete uniform distribution if it takes n distinct integer values, each with equal probability 1/n. The most common example is the score from rolling a fair n-sided die.

For a discrete uniform distribution over values 1, 2, ..., n, the mean and variance can be derived from first principles and should be memorised for S1 (they are not provided in the formula booklet):

- E(X) = \(\frac{n+1}{2}\)
- Var(X) = \(\frac{n^2 - 1}{12}\)

**Worked example:** A fair 8-sided die is rolled, with score X following a discrete uniform distribution over values 1 to 8. Calculate (a) E(X), (b) Var(X), (c) P(X ≤ 5).

1. For n=8, apply the discrete uniform mean formula:
2. $$E(X) = \frac{8 + 1}{2} = 4.5$$
3. Apply the discrete uniform variance formula:
4. $$Var(X) = \frac{8^2 - 1}{12} = \frac{63}{12} = 5.25$$
5. Each value has probability 1/8, so sum probabilities for x=1 to 5:
6. $$P(X \leq 5) = 5 \times \frac{1}{8} = \frac{5}{8}$$

> **tip**
>
> If the discrete uniform distribution does not start at 1 (e.g. values 3,4,5,6), calculate E(X) and Var(X) directly from the general definitions or use coding to shift the range to start at 1 for faster calculation.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Treating non-numerical discrete outcomes as discrete random variables
  - Why it fails: DRVs require numerical values to calculate mean and variance, which cannot be computed for non-numerical data.
  - Correct: Assign numerical values to non-numerical outcomes (e.g. let X=1 for heads, X=0 for tails) before modelling as a DRV.
- **Wrong:** Forgetting to square the coefficient a when calculating Var(aX + b)
  - Why it fails: Variance measures squared deviation from the mean, so scaling X by a scales deviation by a, so variance scales by a².
  - Correct: Always apply Var(aX + b) = a² Var(X), ignoring the constant b entirely.
- **Wrong:** Calculating CDF as P(X < x) instead of P(X ≤ x)
  - Why it fails: The standard Edexcel S1 definition of F(x) is the cumulative probability up to and including the given x value.
  - Correct: Sum all p(x) values for x ≤ the given value when calculating F(x).
- **Wrong:** Using binomial or Poisson distribution formulas for S1 discrete distribution questions
  - Why it fails: Binomial and Poisson distributions are out of scope for S1; only the discrete uniform is a permitted named distribution in this unit.
  - Correct: Derive all probabilities from first principles unless explicitly told the distribution is discrete uniform.
- **Wrong:** Calculating Var(X) as E(X) - [E(X)]² instead of E(X²) - [E(X)]²
  - Why it fails: Missing the square on the X term is a common arithmetic slip that leads to incorrect negative variance values.
  - Correct: Always compute E(X²) (sum of x² multiplied by p(x)) first before subtracting the square of the mean to find variance.

## Cheatsheet

| Concept | Formula / Rule | Exam Note |
| --- | --- | --- |
| Discrete Random Variable | 0 ≤ p(x) ≤ 1, Σp(x) = 1 | Only numerical values qualify |
| CDF F(x₀) | P(X ≤ x₀) = Σ_{x ≤ x₀} p(x) | Non-decreasing, ranges 0 to 1 |
| Expected Value | E(X) = Σ x p(x) | E(aX + b) = aE(X) + b |
| Variance | Var(X) = E(X²) - [E(X)]² | Var(aX + b) = a² Var(X) |
| Discrete Uniform (1 to n) | E(X)=(n+1)/2, Var(X)=(n²-1)/12 | Derive if range does not start at 1 |

## What's next

You have now mastered all discrete random variable content required for Edexcel IAL S1. This topic forms the foundation for all further discrete distribution content in S2, including binomial and Poisson distributions, which are tested heavily in both AS and A2 exams. This topic typically accounts for 8-12% of marks in your S1 exam, so practice past paper questions to build speed and accuracy. Ensure you can apply all linear transformation rules and CDF calculations quickly before moving to more advanced statistics topics.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/edexcel-ial-math-s1-discrete-random-variables/