I'm studying conditional probability and came across this Problem:
Let $(X_o, ... , X_n) \in \mathbb{R}^{n+1}$ be a centered Gaussian vector such that $\mathbb{E}[X_0X_i] = \lambda_i.$
i) What is the distribution of $\mathbb{E}[X_0|X_1,...,X_n]$ ?
ii) Assume $X_1,..,X_n$ are independent, express $\mathbb{E}[X_0|X_1,...,X_n]$ in terms of $\lambda_1,.., \lambda_n$ and $X_1,..,X_n$
I found a pretty similar question here and I think, I have to compute the conditional pdf $f_{X_0|X_1,...,X_n}(x_0|x_1,...,x_n)$ and then compute $ E(X_{0}|X_1,...,X_n)=\int x_{0} f_{X_0|X_1,...,X_n} (x_0|x_1,...,x_n)dx_{0}$
Am I on the right track with this?
Also for the second part, I guess we can use the property $\mathbb{E}[X_0X_i] = \lambda_i$ and the independence of the $X_i$'s to compute the covariance matrix. However, I have a hard time writing these things in a proper way.