Q.If $X$ is a random variable such that $Y = 2X + 3$, and $E(X) = 4$, $Var(X) = 5$, find $E(Y)$ and $Var(Y)$.

Since $Y$ is a linear function of $X$, the standard properties $E(aX+b)=aE(X)+b$ and $Var(aX+b)=a^2Var(X)$ give the answer directly. …

Step 1 — Apply the linearity property of expectation. For $Y=aX+b$ with $a=2$, $b=3$:

Step 2 — Apply the scaling property of variance. Adding a constant ($b=3$) does not change the spread, and scaling by $a=2$ scales the variance by $a^2$:

$Var(Y) = a^2\,Var(X) = 2^2 \times 5 = 4\times5 = 20$ …

Including the constant $b=3$ inside the variance calculation (e.g. writing $Var(Y) = 4\,Var(X) + 3$) — a constant shift never affects variance, only the mean; only the multip …