Law of a Brownian path pinned at both ends

A Brownian motion satisfies $W_0=0$ and is conditioned on $W_t=x$. For $0<s<t$, what is the distribution of $W_s$?

$N\!\left(\tfrac{s}{t}x,\;\tfrac{s(t-s)}{t}\right)$. $(W_s,W_t)$ is bivariate normal with $\operatorname{Cov}=s$ and $\operatorname{Var}(W_t)=t$, so Gaussian conditioning gives mean $\tfrac{s}{t}x$ (linear interpolation) and variance $s-\tfrac{s^2}{t}$. The variance is largest at the midpoint and vanishes at both ends.