$\text{Var}(Y|Z)$ where $Y,Z$ has normal distribution

Question

Let $X_1,...X_{25}$ be a sequence of i.i.d with $N(0,1)$ distribution and let $Y=X_1+...X_{10}$ and $Z=X_1+...+X_{25}$. I have to find $\text{Var}(Y|Z)$. What is the easiest way to do it?

Answer 1

It is easy to see, that $(Y,Z)$ follows the multivariate normal distribution: $$N(\begin{pmatrix}0\\ 0\end{pmatrix} , \begin{pmatrix} 10 & 10 \\ 10 & 25 \end{pmatrix})$$ Using the formula for conditioning in a multivariate normal distribution we get the conditional variance to be $Var(Y|Z) = (1-\rho^2)Var(Y)$, where $\rho$ is the correlation coefficient between $Y$ and $Z$. And since $$Var(Y) = \sum_{i=1}^{10} Var(X_i) = 10$$ and $$\rho = \frac{Cov(Y,Z)}{\sqrt{Var(Y)}\sqrt{Var(Z)}} = \frac{10}{\sqrt{10}\sqrt{25}}=\sqrt{\frac{10}{25}},$$ we get, that $$Var(Y|Z) = (1-\frac{10}{25})10 = 6.$$

(see https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Bivariate_case_2 for reference about the conditional distribution in a bivariate normal distribution)