IB Maths AI 4.10 Notes

Linear transformations

In this section, we study how expectation and variance change when a random variable is transformed, how to work with linear combinations of random variables, and why certain sample statistics are used as unbiased estimates of population parameters.

We begin by reviewing linear transformations, which work the same as introduced in distribution properties.

Let $X$ be a random variable. If we transform $X$ to $aX+b$, then the expectation changes linearly:

$E(aX+b)=aE(X)+b$

This means multiplying by $a$ multiplies the mean by $a$, and adding $b$ shifts the mean by $b$.

The variance changes differently:

$\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)$

The constant $b$ does not affect the variance, because shifting all values by the same amount does not change the spread.

A few examples are:

• If $E(X)=6$, then $E(3X)=3E(X)=18$.
• If $E(X)=4$, then $E(X+10)=E(X)+10=14$.
• If $E(X)=5$, then $E(5-2X)=E(-2X+5)=-2E(X)+5=-10+5=-5$.

For variance:

• If $\mathrm{Var}(X)=5$, then $\mathrm{Var}(3X)=3^2\mathrm{Var}(X)=45$.
• If $\mathrm{Var}(X)=7$, then $\mathrm{Var}(X+10)=\mathrm{Var}(X)=7$.
• If $\mathrm{Var}(X)=4$, then $\mathrm{Var}(5-2X)=(-2)^2\mathrm{Var}(X)=16$.

Standard deviation $\sigma$ is then calculated as the $\sqrt{Var (X)}$.