Prove that for any constant k, extVar(kX + Y) = k2 extVar(X) + extVar(Y) given X and Y are two independent random variables.

Here's the solution to question 3:

Part (i): Prove that for any constant $k$, $\text{Var}(kX + Y) = k^2\text{Var}(X) + \text{Var}(Y)$ given $X$ and $Y$ are two independent random variables.

Step 1: Use the definition of variance, $\text{Var}(Z) = E(Z^2) - (E(Z))^2$.
Let $Z = kX + Y$.

Step 2: Calculate $E(Z)$.
Since expectation is linear:

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