Hello,

If you just have to prove that the mean is 0 and the variance is 1 :

Then use the fact that

Use the fact that

-------------------------------------------------------------

If you want to prove that Z has a normal distribution with mean 0 and variance 1 :

Have you ever dealt with characteristic functions ?

They characterise a distribution.

And for a normal distribution,

So

But

Thus

and this is the characteristic function for a normal distribution with mu=0 and sigma=1.

That was a little extra