Three questions about the expectation of Brownian Motion.

Question

Let W(r) is a standard brownian motion which follows N(0,r).

Then, how to calculate

(1) $E(\int^1_0W(r)^3dr)$

(2) $E(\int^1_0W(r)dr)^2$

(3) $E(W(1)\int^1_0W(r)dr)^2$

For (1), my solution is $E(\int^1_0W(r)^3dr)$ = $\int^1_0E(W(r)^3)dr$ = $\int^1_00dr$ = 0.

But, I cannot get idea about others.

Thanks for your time and consideration.

Answer 1

Hints: $E(\int_0^{1}W(r)dr)^{2}=E\int_0^{1}\int_0^{1}W(r)W(s)dr ds=\int_0^{1}\int_0^{1}\min\{r,s\} drds$ and a similar method gives 3): compute $EW(1)^{2}W(r)W(s)$ and the integrate.