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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.

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    $\begingroup$ Your calculation is correct. I don't understand your question "I wonder..." though. $\endgroup$ – Cave Johnson Jul 29 '18 at 13:19
  • $\begingroup$ Thank you very much. I really appreciate your help a lot. I think the "I wonder" is just the pre-quiz self doubt nerves kicking in. I realize my question is probably quite basic and the rules are quite strict here. Would you recommend that I keep or delete the question? $\endgroup$ – Yes Jul 29 '18 at 13:26
  • $\begingroup$ You are welcome. The question if of good quality and it's absolutely ok if you keep it. $\endgroup$ – Cave Johnson Jul 29 '18 at 14:08

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