Let $\{W'_t\}_{t\in [0,1]}$ be the Brownian motion on the real line obtained by taking the standard Brownian motion $\{W_t\}_{t\ge 0}$ and conditioning on the events $W_1 = 1$ and $0\le W_t\le 1$ for $t\in[0,1]$. My first question is whether $W'$ has a name.

Let $T$ be chosen uniformly at random in $[0,1]$. Secondly, is it possible to calculate explicitly the p.d.f. of $W'_T$?. In words, I am wondering how much does $W'$ favor the middle parts of $[0,1]$ and whether this can be calculated explicitly.