Linear combinations of random variables
Linear combinations
- For a single variable changed by a linear rule:
$$E(aX + b) = aE(X) + b, \qquad \text{Var}(aX + b) = a^2\,\text{Var}(X)$$
- Note the $b$ vanishes from the variance, and the $a$ is squared.
Practice
X has mean 5. What is E(3X − 1)?
E(3X − 1) = 3E(X) − 1 = 3×5 − 1 = 14.
Practice
X has variance 4. What is Var(3X − 1)?
Var(3X − 1) = 3²×Var(X) = 9×4 = 36 (the −1 has no effect).
Practice
In Var(aX + b), the constant b:
Adding a constant shifts the data but does not change its spread, so b drops out.
Two independent variables
$$E(aX + bY) = aE(X) + bE(Y)$$
$$\text{Var}(aX + bY) = a^2\,\text{Var}(X) + b^2\,\text{Var}(Y)$$
- If $X$ is normal, so is $aX + b$; the sum of independent Poissons is Poisson.
You've got it
Key idea
- $E(aX+b) = aE(X) + b$; $\text{Var}(aX+b) = a^2\,\text{Var}(X)$ (the $b$ drops, $a$ is squared)
- independent: $\text{Var}(aX+bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$
- normal stays normal under a linear rule; sums of independent Poissons are Poisson