Metrics Lesson 1

Linearity of Expectation: E[aX + bY + c]

a*E[X] + b*E[Y] + c (Constants pull out cleanly, sums split)

Definition and formula for Var(X)

Var(X) = E[(X - E[X])²] = E[X²] - (E[X])²

Transformation of Variance: Var(a + bX)

b² * Var(X) (Constant shifts vanish; multipliers square)

Variance of a Sum: Var(X + Y)

Var(X) + Var(Y) + 2*Cov(X, Y)

Definition of Covariance Cov(X, Y)

Cov(X, Y) = E[(X - E[X]) * (Y - E[Y])] (Measures linear association)

Covariance rules: Cov(X, X), Cov(X, c), and Cov(a+bX, c+dY)

Cov(X, X) = Var(X) | Cov(X, c) = 0 | Cov(a+bX, c+dY) = b*d * Cov(X, Y)

Correlation formula Corr(X, Y) and its range

Corr(X, Y) = Cov(X, Y) / [SD(X) * SD(Y)], bounded in [-1, 1]

Does Cov(X, Y) = 0 mean independence?

NO. Independence implies Cov = 0, but Cov = 0 only means no linear relation (e.g., Y = X² centered at 0)

What is E[Y | X] and why is it a random variable?

The average outcome of Y given X = x. It is a function of the random variable X, making it a random variable itself.

Pulling Out Known Information: E[f(X) * Y | X]

f(X) * E[Y | X] (Once conditioned on X, f(X) acts as a known constant)

Law of Iterated Expectations (LIE) formula and intuition

E[E[Y | X]] = E[Y] (The weighted average of subgroup averages equals the grand total average)

Law of Total Variance formula and components

Var(Y) = Var(E[Y | X]) + E[Var(Y | X)] (Between-group variance + average within-group variance)

Standardization: If X ~ N(μ, σ²), what is Z and F(x)?

Z = (X - μ) / σ ~ N(0, 1), and F(x) = Φ((x - μ) / σ)

When is the linear combination a + bX + cY normal?

Only when X and Y are JOINTLY normal (marginal normality alone is NOT sufficient)

Estimator unbiasedness definition and Bias formula

Unbiased if E[θ_hat] = θ. Bias = E[θ_hat] - θ

Sample mean properties: Var(X̄_n) and SE(X̄_n)

Var(X̄_n) = σ² / n, and SE(X̄_n) = σ / √n (Spread shrinks at rate √n)

Two-sided t-test statistic τ and 5% rejection rule

τ = |β_hat - β_0| / SE. Reject H0 if τ > 1.96; fail to reject if τ ≤ 1.96

95% Confidence Interval formula and interpretation

[β_hat - 1.96*SE, β_hat + 1.96*SE]. Ex-ante property: 95% of computed intervals across repeated samples contain true fixed parameter