Linear combinations of random variables

The mathematics of combining

• If you know the mean and variance 方差 of two separate processes, what happens when you add them together?
• Linear combinations 线性组合 let you predict the behaviour of combined random variables 随机变量 — essential for quality control, finance, and engineering.

Linear transformations 线性变换 of a single variable

• For a single variable changed by a linear rule:

The Poisson distribution Po(3); its mean and variance both equal lambda

Two independent 独立 variables

• Note: variances always add (even for $X - Y$), because $(-1)^2 = 1$.

Distributions under linear combinations

• If $X$ is normal, so is $aX + b$; the sum of independent normals is normal.
• The sum of independent Poissons is Poisson: if $X \sim \text{Po}(\lambda_1)$ and $Y \sim \text{Po}(\lambda_2)$, then $X + Y \sim \text{Po}(\lambda_1 + \lambda_2)$.

Worked example — combining normals

• $X \sim N(10, 4)$ and $Y \sim N(20, 9)$, independent. Find the distribution of $X + Y$.
• $E(X + Y) = 10 + 20 = 30$. $\text{Var}(X + Y) = 4 + 9 = 13$.
• So $X + Y \sim N(30, 13)$.
• Linear combinations of normal distributions are normal; sums of independent Poisson distributions are Poisson.

You've got it

Write from an expectation of 7 and a variance of 12

• E(X) = 4 and Var(X) = 3. Y = 2X - 1.
• Find E(Y) and Var(Y).