2.8 Mean and Standard Deviation of Random Variables

Mean (Expected Value) of a Random Variable

The mean of a discrete random variable, also called the expected value E(X), is the long-run average outcome if a chance process is repeated many times. Calculate it by multiplying each possible value by its probability and summing: E(X) = Σ xᵢ·P(xᵢ). It's a weighted average, not necessarily an achievable outcome.

Variance and Standard Deviation

Variance measures the spread of a random variable's values around its mean: Var(X) = Σ (xᵢ − μ)²·P(xᵢ). The standard deviation is the square root of variance, σ = √Var(X), and is easier to interpret since it's in the same units as X. Larger σ means outcomes vary more from the mean.

Combining Random Variables

For sums or differences of random variables, means always add/subtract directly: μ(X±Y) = μ(X) ± μ(Y). If X and Y are independent, variances always ADD, even for a difference: Var(X±Y) = Var(X) + Var(Y). Independence is required to add variances, since dependence changes spread.

Key terms

Expected Value E(X)

Σ xᵢ·P(xᵢ); the long-run average outcome of a random variable.

Variance of a Random Variable

Var(X) = Σ (xᵢ − μ)²·P(xᵢ); measures average squared deviation from the mean.

Standard Deviation of a Random Variable

σ = √Var(X); measures typical spread of outcomes from the mean, in original units.

Rule for combining means

μ(X ± Y) = μ(X) ± μ(Y), true whether or not X and Y are independent.

Rule for combining variances (independent variables)

Var(X ± Y) = Var(X) + Var(Y); variances always add when X and Y are independent.

Try an AP-style question

What is E(X)?

1. 2
2. 1.5
3. 1.7
4. 0.25