Q.If $E(X) = 5$ and $Var(X) = 4$ for a random variable $X$, find

Both parts use the standard properties of expectation and variance under a linear transformation, applied directly to the given values.

(i) Finding $E(2X+3)$:

Using the property $E(aX+b) = aE(X)+b$ with $a=2$, $b=3$:

(ii) Finding $Var(3X-2)$:

Using the property $Var(aX+b) = a^2 Var(X)$ with $a=3$, $b=-2$ (the constant $-2$ does not affect variance at all, since shifting a distribution does not change its spread):

Cross-check (conceptual): since $Var(X)=4$ means $SD(X)=\sqrt4=2$, the standard deviation of $3X-2$ should be $|3|\times SD(X) = 3\times2=6$, so $Var(3X-2)$ should equal $6^2=36$ — exactly matching the value obtained directly via the variance property.

Part (ii) can also be verified via the standard-deviation route shown in the cross-check: find $SD(3X-2)=|a|\,SD(X)$ first, then square it to get the variance — both routes must agree.

A very common mistake is applying $Var(aX+b)=aVar(X)+b$ (treating variance like expectation) — the constant $b$ must be dropped entirely (it never affects variance) and the coefficient $a$ must be squared, not used as-is; forgetting to square $a$ is the single most frequent error with this property.