Expectation and Variance of Linear Combinations of Random Variables (PS2 §6.2)

Core Concept

Single variable: the linear transformation $aX + b$

If $X$ is a random variable with mean $\mathrm{E}(X)$ and variance $\mathrm{Var}(X)$, and $a$, $b$ are constants, then:

• Multiplying by $a$ scales both the mean and the spread.
• Adding $b$ shifts the mean but does not affect the spread.

The variance depends only on how spread out the distribution is; shifting every value by the same constant $b$ leaves all distances between values unchanged, so variance is unaffected by $b$.

Two independent variables: the linear combination $aX + bY$

If $X$ and $Y$ are independent random variables, the expectation of any linear combination is found by applying the rule to each term separately. For variance, the independence condition is critical: it ensures there is no covariance term, so variances simply add (after squaring the coefficients).

Key point: The variance formula $\mathrm{Var}(aX + bY) = a^2\mathrm{Var}(X) + b^2\mathrm{Var}(Y)$ holds only when $X$ and $Y$ are independent. This condition is always stated or implied in 9709 questions.