TITLE: Expected value of average of Brownian motion
QUESTION [10 upvotes]: For a standard one-dimensional Brownian motion $W(t)$, calculate: 
$$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$
Note: I am not able to figure out how to approach this problem. All i can think of is that the term $\frac{1}{T}\int\limits_0^TW_t\,dt$ is like 'average'. But not sure how to proceed ahead. I'm relatively new to Brownian motion. I tried searching the forum for some hints..but could not find one. I will really appreciate if you could please guide me in the right direction. Thanks!

REPLY [6 votes]: Another approach would be to show that the random variable
$$\omega \mapsto \int_0^T W_t(\omega) \, dt$$
is a centered normal random variable with variance $\int_0^T (s-T)^2 \, ds=\frac{T^3}{3}$. You can find a proof here. (The proof is a bit lenghthy, but you don't need Itô-Calculus to prove it.)