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CIE A-Level Maths Study Notes

4.4.1 Probability Distributions of Discrete Random Variables

A solid grasp of probability distributions of discrete random variables is essential. This detailed exploration covers the creation of probability distribution tables, the calculation of expected value E(X) E(X) and variance Var(X) Var(X), accompanied by practical examples. These concepts are pivotal in understanding statistical analysis and data interpretation.

Probability Distributions

  • A probability distribution describes how likely different outcomes are in an experiment.
  • For discrete random variables, outcomes are distinct, like countable numbers.

Discrete Random Variables

  • These variables take specific values, often whole numbers.
  • Examples: Number of correct answers on a test, number of heads in coin flips.

Probability Distribution Tables

  • These tables show probabilities for each outcome of a discrete random variable.

Example: Coin Toss

  • Experiment: Tossing a fair coin twice.
  • Random variable X = number of heads.
  • Possible X values: 0, 1, 2.
  • Each outcome (head or tail) is equally likely.
probability distribution

Expected Value (E(X))

  • Represents the 'average' outcome of a random variable.
  • Calculated as: E(X) = Sum of [x * P(x)].
  • Example: Coin Toss, E(X) = 0 0.25 + 1 0.50 + 2 * 0.25 = 1.
  • In two coin tosses, expect 1 head on average.
mean of probability distribution

Variance (Var(X))

  • Measures how spread out the data is.
  • Calculated as: Var(X)=E(XE(X))2Var(X) = E (X - E(X))^2.
  • Example: Coin Toss, Var(X)=0.25(01)2+0.50(11)2+0.25(21)2=0.5Var(X) = 0.25 (0 - 1)^2 + 0.50 (1 - 1)^2 + 0.25 * (2 - 1)^2 = 0.5.
  • Variance for number of heads in two tosses is 0.5.
variance of probability distribution
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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