September 24

Linear functions of r.v.’s

A linear function of a random variable is a new random variable created by a linear transformation of an existing one. If $X$ is a random variable, a linear function of $X$ is defined as $Y = aX + b$, where $a$ and $b$ are constants. This transformation shifts and scales the values of $X$.

The key properties of these new random variables relate to their expected value and variance.

Expected Value

The expected value of a linear function of a random variable can be calculated using the linearity of expectation.

• $E(aX + b) = a \cdot E(X) + b$

This means that the expected value of the transformed variable is simply the transformed expected value.

• Example: If a random variable $X$ has an expected value of $E(X) = 10$, and you define a new variable $Y = 3X - 5$, the expected value of $Y$ is: $E(Y) = E(3X - 5) = 3 \cdot E(X) - 5 = 3(10) - 5 = 25$

Variance

The variance of a linear function of a random variable is affected differently by the constants $a$ and $b$.

• $Var(aX + b) = a^2 \cdot Var(X)$

Note that the constant $b$ does not affect the variance. This is because adding a constant to every value in a distribution simply shifts the entire distribution without changing its spread or shape. The variance measures the spread, so it remains unchanged.

• Example: If a random variable $X$ has a variance of $Var(X) = 4$, and you define a new variable $Y = 3X - 5$, the variance of $Y$ is: $Var(Y) = Var(3X - 5) = 3^2 \cdot Var(X) = 9 \cdot 4 = 36$

Summary

Property	Formula
Expected Value	$E(aX + b) = a \cdot E(X) + b$
Variance	$Var(aX + b) = a^2 \cdot Var(X)$