Let $\displaystyle {B}(t)$ be a Brownian motion and $\displaystyle 0\leq{s}<{t}.$ Show that the conditional distribution of $\displaystyle {B}(s)$ given $\displaystyle {B}(t)=b$ is Normal and give its mean and variance.

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- Jun 10th 2009, 02:04 AMsymmetry7Still more Brownian motion
Let $\displaystyle {B}(t)$ be a Brownian motion and $\displaystyle 0\leq{s}<{t}.$ Show that the conditional distribution of $\displaystyle {B}(s)$ given $\displaystyle {B}(t)=b$ is Normal and give its mean and variance.