Given that $\{B_t,t\ge0\}$ is a standard Brownian process. What is the conditional distribution of $B(s)$ given $B(t_1)=x_1$ and $B(t_2)=x_2$, for $0
My try: First i tried to write $B(s)=B(s)-B(t_1)+B(t_1)$ and then try to find the conditional distribution $\Pr(B(s),B(t_1)=x_1,B(t_2)=x_2)$ then we get $P(B(s)-x_1+x_1,B(t_1)=x_1,B(t_2)=x_2)$-but then i have no idea how to proceed. I am thinking of a 2nd apporach, using the moment generating function but seems that it is not easy to get the moment generating function of a condition random variable, as i don't even know how its distributed.