AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators

Use $E(aX+b)=aE(X)+b$ with $a=5$, $b=10$.

$E(5X+10)=5(40)+10=200+10$.

$E(5X+10)=210$.

$200$ omits the constant $b=10$, forgetting it shifts the mean.

$50$ mistakenly computes $a+b=5+10=15$... actually this comes from adding $a$ and $b$ directly to $E(X)$ incorrectly, e.g. $40+10=50$, ignoring the scaling by $a$.

$80$ results from computing $2\times E(X)$ or similar arithmetic slip, not applying the correct transformation.

Use $\text{Var}(aX+b)=a^2\text{Var}(X)$ with $a=4$.

$\text{Var}(4X-7)=4^2\times 25=16\times 25$.

$\text{Var}(4X-7)=400$.

$393$ incorrectly subtracts the constant $7$ from the variance, e.g. $400-7$, treating $b$ as if it affects spread.

$100$ uses $a$ instead of $a^2$, i.e. $4\times 25$.

$16$ is simply $a^2$ alone, forgetting to multiply by $\text{Var}(X)$.