Variance of Expectation Question

Question

I'm trying to figure out $\mathrm{Var}(\mathbb{E}(M|N=k))$, where $(M|N=k)$ is $\sum_{i=1}^k 10 - s_i$ where all the $s_i$'s are uniformly distributed on $(0,10)$. My own calculation gives $0$ as the answer, so I'm wondering if I'm doing something wrong.

If I'm not mistaken, $\mathbb{E}(M|N=k) = 5k$ and $\mathrm{Var}(\mathbb{E}(M|N=k))$ yields $(5k)^2 - (5k)^2 = 0$.

Any help would be appreciated. Thanks.

Answer 1

If $k$ is non-random, then the expectation is deterministic and its variance is, indeed, 0.