Recipe

Find the mean and variance after a linear transformation

Use when: you have $Y=aX+b$ and need $E(Y)$ , $\operatorname{Var}(Y)$ , or a missing constant, from what's known about $X$

Goal: the complete pair $E(Y)$ and $\operatorname{Var}(Y)$ (or a missing constant), using the linear-transformation rules for mean and variance together

Taught in Variance under a linear transformation · extends Find the mean after a linear transformation

Quick recipe

Identify $a$ , $b$ , and whatever's known about $X$
Decide which quantity you need: $E(Y)$ , $\operatorname{Var}(Y)$ , or a missing constant
Apply $\operatorname{Var}(Y)=a^2\operatorname{Var}(X)$ when variance is what's asked

1
Write the transformation as $Y=aX+b$ , and read off $a$ and $b$ ; also note whatever's given about $X$ , its mean, its variance, or both, depending on what the problem asks for.

Example

$X$ has $\operatorname{Var}(X)=4$ ; $Y=-3X+7$ gives $a=-3$ and $b=7$ .

Why: every later step is just substituting these numbers into one of two formulas, so misreading $a$ or $b$ propagates through the whole problem.
2
Read the problem again and pin down exactly what it's asking for before touching either formula.
- $E(Y)$ is unknown — apply $E(Y)=aE(X)+b$ , the same as Step 3 of the mean recipe above.
- $\operatorname{Var}(Y)$ is unknown — apply $\operatorname{Var}(Y)=a^2\operatorname{Var}(X)$ in Step 3 below, noting $b$ drops out entirely.
- a target mean is given and $a$ or $b$ is the unknown — solve $E(Y)=aE(X)+b$ for that constant, exactly as in the mean recipe's backward direction.
Why: the mean and variance formulas both use $a$ and $b$ , but they use them differently, so misreading which quantity is asked for leads to applying the wrong one.
3
Square $a$ first, the sign disappears once squared, then multiply by $\operatorname{Var}(X)$ . The shift $b$ never appears anywhere in this formula.

Example

$\operatorname{Var}(Y) = (-3)^2(4) = 9(4) = 36$

⚡ Shortcut

If the transformation is a pure shift, $a=1$ and only $b$ changes, then $\operatorname{Var}(Y)=\operatorname{Var}(X)$ automatically, since $1^2=1$ . No multiplication is needed at all: $Y=X+100$ has exactly the same variance as $X$ .

Why: squaring $a$ is why the SIGN of $a$ never matters for spread, only its magnitude, unlike the mean formula, where the sign of both $a$ and $b$ matter completely.

Common mistakes

Forgetting to square $a$ , using $\operatorname{Var}(Y)=a\operatorname{Var}(X)$ instead of $a^2\operatorname{Var}(X)$ .
Including $b$ in the variance computation at all; a pure shift never changes spread.
Applying the variance formula when the problem actually asked for the mean, or the mean formula when it asked for the variance.
Forgetting to add $b$ , or losing its sign, when the problem actually needs the mean recipe's Step 3 instead.