I need to find , where is a standard Brownian motion.

I tried the following.

Brownian motion has independent increments, and I should be able to write the above sum as two independent standard normal variables, i.e.

, where and are independet standard normal variables.

Thus...

In the last line I used the fact that independent random variables have factoring moment-generating function and also the fact that .

Is the above reasoning correct, if not, what is wrong? Thanks!