The Expectation of $e^{W_{t}+W_{s}} $ for fixed t and s

Question

Here $W_{t}$ is brownian motion or the Wiener process. I am actually trying to find the probability distribution of $W_{t}+W_{s} $ for fixed t and s by exploiting Levy's continuity theorem. I know that $W_{t}-W_{s} $ is independent of $W_{s} $ for t>s but I don't know how I can justify that $e^{W_{t}-W_{s}} $ is independent of $e^{W_{s} } $ which would allow me to calculate this Expectation value. Also if this is true then I get a probability distribution with variance t+3s for t>s is there any intuition behind this?

Answer 1

Answer for the first part: If $X$ and $Y$ are independent and $f$ is any measurable function $\mathbb R \to \mathbb R$ then $f(X)$ and $f(Y)$ are independent. Since $f(x)=e^{x}$ is continuous it is Borel measurable,