Difference of normal random variables

Question

I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that... $$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$ I understand that the variance is going to be Var($X_{s+t})$ + Var($X_{s})$ + something to do with co variance of the two - but how do I calculate this?

Thanks for the help in advance!

Answer 1

The formula is $$\operatorname{Var}(X_{s+t} - X_s) = \operatorname{Var}(X_{s+t}) + \operatorname{Var}(-X_s) + 2 \operatorname{Cov}(X_{s+t}, -X_s) = s + t + s - 2 \operatorname{Cov}(X_{s+t}, X_s).$$ Presumably, $\operatorname{Cov}(X_{s+t}, X_s) = \sqrt{s(s+t)}$, but we cannot deduce this from what you have written in your question. With the context that you have omitted, it may be possible to compute that covariance.