I missed a lecture the other day on Brownian motion and am writing a class quiz on Monday. I have been working through some questions but seem to struggle with this one. Apparently it was discussed in the missed lecture.
We have $X$ being standard Brownian motion. Give the distribution of $X(2)$ given that $X(1) = 10$.
So my attempt at it is that I assume the Markov property of Brownian motion. I then get
$$ \left[X(2) \in x \mid X(1) = 10 \right] = p(x-10,2-1) $$ From this we assume: $$\left[X(2) \in x \mid X(1) = 10 \right] \sim N(10,1) $$
On the other hand, I wonder whether all X(t) are not independent. I know the increments are independent. Independence would imply that the condition does not affect the distribution though.
Could I please ask whether what I have done seems correct and for you to please guide me in amending what is wrong. Thank you for your time.