Hello everyone!

The question I have is:

$\displaystyle \xi$ and $\displaystyle \eta$ are indpt standard normal random variables, we have a random process $\displaystyle X_t$ ,where:

$\displaystyle X_t=(\xi-\eta)\sqrt{t}, t\geq0$

I would like to find out the distribution (one dimension FDD) of $\displaystyle X_t$ ,for a fixed $\displaystyle t>0$

Here is what I did:

$\displaystyle F_t(x)=P((\xi-\eta)\sqrt{t}\leq x)$

$\displaystyle =\int P(\xi\leq\frac{x}{\sqrt{t}}+s|\eta=s)dF_\eta(s)$ by Total Probability Formula

$\displaystyle =\int P(\xi\leq\frac{x}{\sqrt{t}}+s)dF_\eta(s)$ by independence

$\displaystyle =\int F_\xi(\frac{x}{\sqrt{t}}+s)\cdot\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2} s^2} ds$

here is where I stuck...how should I deal with $\displaystyle F_\xi(\frac{x}{\sqrt{t}}+s)$, or is there other ways to this kind of things?

What about n dimensional FDD?

Could anyone help me? :)