Chapter 3 – Linear Combinations of Random Variables

Vocabulary cards covering the main definitions, rules and key points for expectation, variance, and distributions when forming linear combinations of random variables.

Linear Combination (of random variables)

An expression of the form aX + bY + … where a, b,… are constants and X, Y,… are random variables.

Expectation (Mean)

The long-run average value of a random variable; denoted E(X) or μ.

Variance

A measure of spread; the expected squared deviation from the mean, Var(X) = E[(X−μ)²].

Shift Rule for the Mean

E(X + b) = E(X) + b for any constant b.

Scale Rule for the Mean

E(aX) = a·E(X) for any constant a.

General Mean Transformation

E(aX + b) = a·E(X) + b.

Shift Rule for the Variance

Adding a constant does not change variance: Var(X + b) = Var(X).

Scale Rule for the Variance

Var(aX) = a²·Var(X).

General Variance Transformation

Var(aX + b) = a²·Var(X).

Sum – Mean Property

For independent (or any) X, Y: E(X + Y) = E(X) + E(Y).

Sum – Variance Property

For independent X, Y: Var(X + Y) = Var(X) + Var(Y).

Difference – Variance Property

For independent X, Y: Var(X − Y) = Var(X) + Var(Y).

Linear Combination of Two Independent Variables

E(aX + bY) = aE(X) + bE(Y) and Var(aX + bY) = a²Var(X) + b²Var(Y).

Closure of the Normal Distribution

If X and Y are independent normals, then aX + bY is also normal.

Standardisation (Z-score)

Z = (X − μ)/σ, transforming X ~ N(μ,σ²) to Z ~ N(0,1).

Binomial Mean

For X ~ B(n,p): E(X) = np.

Binomial Variance

For X ~ B(n,p): Var(X) = np(1−p).

Poisson Distribution

Discrete distribution with P(X=k)=e^{−λ}λ^k/k!, mean and variance both equal λ.

Sum of Independent Poissons

If X ~ Po(λ) and Y ~ Po(μ), then X + Y ~ Po(λ + μ).

Independent Random Variables

Variables whose outcomes do not influence each other; knowing one gives no information about the other.

2X vs X₁ + X₂

2X doubles a single observation; X₁+X₂ adds two independent observations, changing variance.

Key Point 3.1 (Additive Constant)

E(X + b) = E(X) + b and Var(X + b) = Var(X).

Key Point 3.2 (Scaling)

E(aX) = aE(X) and Var(aX) = a²Var(X).

Key Point 3.3 (Full Transformation)

E(aX + b) = aE(X) + b ; Var(aX + b) = a²Var(X).

Key Point 3.4 (Sum of Independent Variables)

E(X + Y) = E(X) + E(Y); Var(X + Y) = Var(X) + Var(Y).

Key Point 3.5 (General Linear Combination)

For independent X,Y: E(aX + bY) = aE(X)+bE(Y); Var(aX + bY)=a²Var(X)+b²Var(Y).

Key Point 3.6 (Normal Linear Combo)

aX + bY is normal if X and Y are independent normals.

Key Point 3.7 (Poisson Sum)

The sum of independent Poisson variables is Poisson with parameter equal to the sum of parameters.

Standard Deviation

The square root of variance; measures average distance from the mean.