I need to find $\displaystyle \mathbb{E}\left[e^{W(1)+W(2)}\right]$ , where $\displaystyle W$ is a standard Brownian motion.

I tried the following.

$\displaystyle W(1)+W(2)=(W(2)-W(1))+2W(1) $

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

$\displaystyle (W(2)-W(1))+2W(1)=G_{1}+2G_{2} $, where $\displaystyle G_1 $ and $\displaystyle G_2$ are independet standard normal variables.

Thus...

$\displaystyle \mathbb{E}\left[e^{W(1)+W(2)}\right]= \mathbb{E}\left[e^{G_{1}+2G_2}\right]=\mathbb{E}\left[e^{G_1}\right] \cdot \mathbb{E}\left[e^{2G_2}\right] = e^{1/2} \cdot e^{2} = e^{\frac{5}{2}} $

In the last line I used the fact that independent random variables have factoring moment-generating function and also the fact that $\displaystyle \mathbb{E}\left[e^{uG}\right] = e^{\frac{1}{2}u^2} $.

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