Let $Y(t)=[B(t)|B(s)=c,B(u)=d]$ for $t \in[s,u]$ with $0<s<u$ and $c,d$ given. What is the distribution of $Y(t)$ for any $t \in [s,u]$? What is $E[Y(t)]$ and $Var[Y(t)]$?
I know that the marginal distributions of Brownian motion are normal and that the Variance is equal to length of time, however I'm not sure how to approach the distribution of Brownian motion conditional on another.