How Does a Linear Transformation Affect E(X) and Var(X)?

For a linear transformation $Y=aX+b$, expected value becomes $\mathrm{E}(Y)=a\mathrm{E}(X)+b$, while variance becomes $\mathrm{Var}(Y)=a^2\mathrm{Var}(X)$. The constant $b$ shifts the centre of the distribution but does not change its spread.

Why These Rules Work

The expected value is the long-run mean of a random variable. Multiplying every possible value of $X$ by $a$ multiplies the mean by $a$, while adding $b$ to every value adds $b$ to the mean. Therefore,

$$\mathrm{E}(aX+b)=a\mathrm{E}(X)+b.$$

The variance measures the expected squared distance of values from their mean. First consider the deviation of $Y$ from its own expected value:

$$Y-\mathrm{E}(Y)=aX+b-\left(a\mathrm{E}(X)+b\right)=a\left(X-\mathrm{E}(X)\right).$$

The translation $b$ cancels because it moves every value and the mean by the same amount. Squaring the remaining deviation produces the factor $a^2$, giving

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

Since standard deviation is the positive square root of variance, the standard deviation becomes $\sigma_Y=|a|\sigma_X$. The modulus is necessary because standard deviation cannot be negative.

Two special cases help check the formulas. If $a=0$, then $Y=b$ is constant, so $\mathrm{E}(Y)=b$ and $\mathrm{Var}(Y)=0$. If , the distribution is reflected and shifted: its centre changes according to , but its variance remains . These checks are useful for detecting sign or scaling errors.

Worked Example

Suppose $\mathrm{E}(X)=10$, $\mathrm{Var}(X)=9$, and $Y=-2X+5$. Then

$$\mathrm{E}(Y)=-2(10)+5=-15,$$

and

$$\mathrm{Var}(Y)=(-2)^2(9)=36.$$

Also, $\sigma_X=3$, so $\sigma_Y=|-2|(3)=6$, consistent with .

A common misconception is that a negative multiplier makes the variance negative. This is impossible because variance is non-negative: the multiplier is squared. Another mistake is adding $b$ to the variance, even though translation does not alter distances from the mean.

IB Exam Technique

This is IB Mathematics AA HL, AHL 4.14: Properties of discrete and continuous random variables. In a “show” question, derive the variance rule using deviations from the mean. In a “calculate” question, state the relevant formula before substituting values, and distinguish clearly between variance and standard deviation.