How Is the Distribution of B_s Given B_t Computed in Brownian Motion?

[IniquiTrance]

I computed the distribution of $B_s$ given $B_t$, where $0\leq s <t$ and $\left\{B_t\right\}_{t\geq 0}$ is a standard brownian motion. It's normal obviously..

My question is, how do I phrase what I've done exactly? Is it that I computed the distribution of $B_s$ over $\sigma(B_t)$?

 

[chiro]

Hey IniquiTrance.

If you partition the distributions so that they don't overlap then you can use the properties of a Wiener (or Brownian motion) process and that should be enough in terms of the justification used.