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.