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Discrete Random Variables

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Edexcel International A Level

Mathematics

Statistics 1

Discrete Random Variables

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Table of Contents

1. Discrete Random Variables

What is a Discrete Random Variable?

Definition: A random variable is a variable whose value depends on the outcome of a random event. A discrete random variable (DRV) can only take certain values within a set.
Key Characteristics
  • Values depend on random outcomes (unpredictable until event occurs)
  • Denoted using uppercase letters (X, Y, Z)
  • Specific outcomes denoted with lowercase letters (x, y, z)
  • Can only take specific isolated values (not continuous)
  • Usually count something
  • Can have finite or infinite number of values
Notation: P(X = x) means "the probability that random variable X takes the value x"
Examples of Discrete Random Variables
Example Possible Values Type
Number of heads in 20 coin flips 0, 1, 2, ..., 20 Finite
Number of emails received per hour 0, 1, 2, 3, ... Infinite
Number of dice rolls until a 6 appears 1, 2, 3, ... Infinite
Bingo ball number drawn 1, 2, 3, ..., 90 Finite
Tip: When identifying a discrete random variable, ask yourself: "What am I counting?" This helps distinguish discrete from continuous variables.

2. Discrete Probability Distributions

What is a Probability Distribution?

Definition: A discrete probability distribution fully describes all values that a discrete random variable can take, along with their associated probabilities. It can be presented as a table or function.
Key Properties
  • Can be shown in a table format (like GCSE)
  • Can be given as a function (probability mass function)
  • Can be represented as a vertical line graph
  • All probabilities must sum to 1: ΣP(X = x) = 1
  • Each probability must be between 0 and 1
Example: Probability Distribution Table

A spinner with sections labeled -2, 0, 1/3, and 5

x -2 0 1/3 5
P(X = x) 1/4 1/8 1/8 1/2

Check: 1/4 + 1/8 + 1/8 + 1/2 = 2/8 + 1/8 + 1/8 + 4/8 = 8/8 = 1 ✓

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Also in the full note
  • 3. Cumulative Probabilities
  • 4. Expected Value E(X) and Variance Var(X)
  • 5. Transformations: aX + b
  • 6. Discrete Uniform Distribution
  • 7. Key Facts Summary
  • 8. Pre-Exam Preparation
  • The Cumulative Distribution Function
  • Calculating Cumulative Probabilities
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More Statistics 1 topics

Statistical Measures
Basic Statistical Measures
Discrete Probability Distributions
Calculating Probabilities & Events
Basic Probability
Conditional Probability
Frequency Tables
Working with Data
Stem & Leaf Diagrams
Correlation & Regression