Finding Conditional Expectation and variance E(Y|X=x)

Question

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x).

Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian process with mean=0 and variance = 1. Here's what I did:-

So, I found the joint probability function of X and Y, i.e., P(X,Y). Divided it by Marginal density function of X, i.e., P(X) and got the P(Y|X). I have the value of ρ, and variance for X & Y = 1 and E(X)=E(Y)=0.

I know x(t) = 0.5 and τ=0.1 now. How do I find E(Y|X=0.5) and Var(Y|X=0.5)?

Any hint or solution? I know to find the conditional expectation I need to multiply the PDF with y and integrate over y. But I really need an example.

Answer 1

Hi if your process is Gaussian then the joint law $(X_t,X_{t+\tau})$ is a multivariate Gaussian random variable. You only need to apply in this case the formulas in wiki link at the section on conditional distribution to get what you want.

Best regards

Answer 2

great to see you are interested in GPs. Someone asked a similar question about how to make predictions on a GP and I think I gave a fairly comprehensive work through with sample data that they gave.

I think seeing it in action will answer your question. Here is the link.