Integral of a Brownian bridge with respect to time Let $(W_s)_{s\geq 0}$ be a Brownian motion and $t$ a fixed point in time. What is the distribution of $$\Big.\int_0^tW_sds\Big|W_t$$
i.e. the integral of a Brownian bridge with respect to time? Is it Gaussian? 

For convenience, I will take $t=1$. The same kind of calculations apply for general $t$. As, $W_s\vert W_1$ is Gaussian process, the lebesgue integral is  Gaussian RV : Approximate the given integral as Riemann sums and each Riemann sum is Gaussian and hence the limit will be Gaussian. Look here for similar problem : click here 
The expectation of it is given by $\int_0^1tW_1dt=0.5W_1$
