Discrete Random Variables: Probability Distributions, Expectation and Variance (9709 S1 §5.4) — Mathematics

Introduction

A discrete random variable (DRV) is a variable whose value is determined by a random experiment and can only take a finite (or countably infinite) set of distinct values. In 9709 Paper 5, questions regularly ask you to construct the full probability distribution, then find the mean and spread of that distribution. These calculations underpin further topics such as the binomial distribution, so mastering them here pays dividends throughout the paper.

Core Concept

What is a probability distribution?

For a discrete random variable $X$ , its probability distribution lists every possible value $x_i$ together with the probability $P(X = x_i)$ that $X$ takes that value. The two fundamental rules (from Addition and Multiplication of Probabilities) ensure:

$P(X = x_i) \geq 0$ for all $i$
$\displaystyle\sum_{\text{all } i} P(X = x_i) = 1$

A probability distribution table is the standard way to display this:

$x$	$x_1$	$x_2$	$\cdots$	$x_n$
$P(X = x)$	$p_1$	$p_2$	$\cdots$	$p_n$

Expectation E(X)

The expectation (or mean) of $X$ is the long-run average value. It is a weighted average of all possible values, weighted by their probabilities:

E(X) = \sum_{\text{all } i} x_i \, p_i

Variance Var(X)

The variance measures the spread of the distribution about its mean. The cleanest formula for calculation is:

\operatorname{Var}(X) = E(X^2) - [E(X)]^2

where $E(X^2) = \displaystyle\sum_{\text{all } i} x_i^2 \, p_i$ .

The standard deviation is $\sigma = \sqrt{\operatorname{Var}(X)}$ .

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