Expected Value and Variance

The expected value E[X] of a random variable X is its probability-weighted average, equal to the long-run mean of many independent realizations. For a discrete random variable taking values x_i with probabilities P(X = x_i), E[X] = Σ x_i P(X = x_i). For a continuous random variable with probability density function f(x), E[X] = ∫_{-∞}^{∞} x f(x) dx (provided the integral converges absolutely). Expectation is linear: E[aX + bY] = a E[X] + b E[Y] even when X and Y are dependent. Variance quantifies the expected squared deviation from the mean μ = E[X]: Var(X) = E[(X − μ)²] = E[X²] − μ². The standard deviation σ = √Var(X) restores the original units. Variance is always non-negative and is zero only for a constant (almost surely). For independent random variables, Var(X + Y) = Var(X) + Var(Y); the cross term vanishes because Cov(X, Y) = 0. Covariance itself is defined by Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)] and measures linear association; its normalized form is the correlation coefficient.