Random variables and their distributions

A random variable is a rule that attaches a real number to every outcome of a random experiment. Formally it is a real-valued function with the sample space as its domain.

Toss a coin twice: S = {HH, HT, TH, TT}. If X is the number of heads, X(HH) = 2, X(HT) = X(TH) = 1, X(TT) = 0.

Several random variables can live on one sample space. On the same S, let Y = (number of heads) − (number of tails): Y(HH) = 2, Y(HT) = Y(TH) = 0, Y(TT) = −2.

The rationalised textbook stops at this remark. The three points below are the JEE extension.

JEE extension: the probability distribution of X lists each value xᵢ with its probability pᵢ = P(X = xᵢ). Every pᵢ ≥ 0 and the pᵢ add to 1. For X above: P(X = 0) = 1/4, P(X = 1) = 1/2, P(X = 2) = 1/4.

JEE extension: the mean (expectation) of X is E(X) = Σ xᵢpᵢ, the probability-weighted average of its values. For X above, E(X) = 0(1/4) + 1(1/2) + 2(1/4) = 1.

JEE extension: the variance is Var(X) = Σ xᵢ²pᵢ − [E(X)]². For X above, Σ xᵢ²pᵢ = 0 + 1/2 + 1 = 3/2, so Var(X) = 3/2 − 1 = 1/2.