Discrete Random Variables

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Draw up a probability distribution table relating to a given situation involving a discrete random variable X, and calculate E(X) and Var(X)
  2. Use formulae for probabilities for the binomial and geometric distributions, and recognise practical situations where these distributions are suitable models
  3. Use formulae for the expectation and variance of the binomial distribution and for the expectation of the geometric distribution

1. What is a Discrete Random Variable?

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A random variable is a quantity whose value depends on chance (random outcomes). We use capital letters like X to represent random variables.

A discrete random variable can only take specific, separate values (usually whole numbers). For example:

  • The number of heads when you flip 5 coins (can be 0, 1, 2, 3, 4, or 5)
  • The score when you roll a die (can be 1, 2, 3, 4, 5, or 6)
  • The number of students absent from class on a given day

Discrete random variables are different from continuous variables (like height or time) which can take any value within a range.

2. Probability Distribution Tables

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A probability distribution shows all the possible values a random variable can take and the probability of each value occurring.

We write this information in a table with two rows:

  • First row: all possible values of X
  • Second row: the probability of each value, written as P(X = x)

Important rules:

  • All probabilities must be between 0 and 1
  • The sum of all probabilities must equal 1 (because one of the outcomes must happen)

Example: Minimum Score of Two Dice

Two fair dice are thrown. Let X be the smaller of the two scores (if the scores are different) or the score on one die (if they are the same).

Step 1: List all possible outcomes. When you throw two dice, there are 6 × 6 = 36 equally likely outcomes.

Step 2: For each possible value of X, count how many outcomes give that value.

  • X = 1: This happens when at least one die shows 1

    • Outcomes: (1,1), (1,2), (2,1), (1,3), (3,1), (1,4), (4,1), (1,5), (5,1), (1,6), (6,1)
    • Count: 11 outcomes

  • X = 2: The minimum is 2 (so both dice show 2 or more, with at least one showing 2)

    • Outcomes: (2,2), (2,3), (3,2), (2,4), (4,2), (2,5), (5,2), (2,6), (6,2)
    • Count: 9 outcomes

  • X = 3: Count = 7 outcomes

  • X = 4: Count = 5 outcomes

  • X = 5: Count = 3 outcomes

  • X = 6: Count = 1 outcome (only (6,6))

Step 3: Calculate probabilities by dividing each count by 36.

The probability distribution table is:

x123456
P(X = x)11/369/367/365/363/361/36

Check: 11/36 + 9/36 + 7/36 + 5/36 + 3/36 + 1/36 = 36/36 = 1 ✓

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