Random Variables

A random variable X takes numerical values determined by chance, for example the score on a die. A discrete random variable takes separate values and is described by a probability distribution P(X = x) with ΣP = 1. A continuous variable takes values on an interval and uses a density curve.

Expectation

The expectation (mean) is the long-run average value.

Variance and standard deviation

The variance measures the spread; the standard deviation has the same units as X.

Worked exampleA fair die

Find E(X) and Var(X) for a fair six-sided die.

Expectation.
E(X²).
Variance.

Answer: E(X) = 3.5, Var(X) = 2.92, σ ≈ 1.71.

All six bars are equal: a uniform distribution.

Distribution	Mean	Variance
Uniform on 1 to n	(n + 1)/2	(n² − 1)/12
Binomial (n, p)	np	npq
Poisson (λ)	λ	λ
Normal (μ, σ²)	μ	σ²

Try itA fair n-sided die

Choose the number of faces. See the mean, variance and standard deviation.

Check yourself For a random variable, the sum of all P(X = x) is

Check yourself If E(X) = 4 then E(3X + 2) =

Check yourself If Var(X) = 5 then Var(2X) =

Key terms

Random variable: A variable whose value depends on chance.
Expectation: The mean value of a random variable.
Variance: The mean of the squared distances from the mean.
Standard deviation: The square root of the variance.

Summary

A discrete probability distribution has ΣP = 1.
E(X) = Σ x P(x).
Var(X) = E(X²) − [E(X)]².
E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X).