Expected Value and Variance of a Random Variable

The expected value is the long-run average of a random variable. It is what you would get if you repeated the process a very large number of times and averaged the results.

Expected Value

E(X) = Σ x × P(x)

• E(X): the expected value, also written μ. It is the distribution's balance point.
• x: each possible value of the random variable.
• P(x): the probability of that value.
• Σ (sigma): add up the product x × P(x) across all possible values.

Variance

Var(X) = Σ (x − μ)² × P(x)

• Var(X): the variance of the random variable, which measures how widely its values spread around the expected value.
• x − μ: the distance of a value from the expected value.
• P(x): each squared distance is weighted by how likely that value is.

The standard deviation is σ = √Var(X), which returns the spread to the original units.

Example: A support team receives a number of urgent tickets in a day, X, with the following probabilities. Find E(X) and Var(X).

1. E(X) = 0(0.1) + 1(0.4) + 2(0.3) + 3(0.2) = 0 + 0.4 + 0.6 + 0.6 = 1.6
2. Var(X) = (0 − 1.6)²(0.1) + (1 − 1.6)²(0.4) + (2 − 1.6)²(0.3) + (3 − 1.6)²(0.2)
3. Var(X) = 0.256 + 0.144 + 0.048 + 0.392 = 0.84
4. σ = √0.84 ≈ 0.92

The team can expect about 1.6 urgent tickets a day, plus or minus about 0.92. You calculate a model’s expected loss from its possible errors the same way.