Assume that $S_n = \sum_{i=1}^n X_i$ with $X_i$ being i.i.d. and $E[X_1] = 25000$ and $V[X_1] = 2500$. Define $Y(t) = 125 \times 10^6t - S_{5000t}$ for $t \in \Bbb N$.
Calculating $E[Y(t)]$ is pretty easy, but I don't see how to calculate $Var[Y(t)]$. I would have started like this:
$Var(125 \times 10^6t - S_{5000t}) = Var(125 \times 10^6t) - Var(S_{5000t}) = 0 - Var(S_{5000t}),$ which obviously yields a negative value.
Where am I mistaken here?
Since $$\mathbf{Var}(a+bX) = b^2\mathbf{Var}(X^2)$$ it follows that \begin{align} \mathbf{Var}(125 \times 10^6t - S_{5000t}) &= (-1)^2\mathbf{Var}(S_{5000t}). \end{align}