$W_t$ is a Wiener process, $W_0 = 0$. Let $Y_t \,\,\text{be}\,\, {W_t | W_T = x}$. What is the distribution of $Y_t$?
I am not so sure, I do get that $W_T - W_t = x - W_t$ would depend on $(T-t)$ and without the condition $W_t$ would have a normal distribution, but I have no idea how to work around this condition on $Y_t$ and how it would affect $Y_t$ distributions. Any help?