integral of square of Brownian motion

Question

What is expectation of $$\int_0^t B(s)^2ds$$ where $B(s) is standard Brownian motion. Is the integral a well known random variable?

Answer 1

The expectations is easy to calculate, using Fubini's theorem, which applies since the integrand is positive: $$ \begin{align} E\left[\int_0^t B(s)^2 ds\right] = \int_0^tE[B(s)^2]ds = \int_0^ts\,ds = \frac{t^2}{2} \end{align} $$