Poisson Conditional Probability Question

JHoop

New Member
#1
Hey Guys

I've been racking my brain for ages trying to solve this but I can't seem to get the last step.

The question is, given
Code:
 0<u<t , 0<=k<=n
Show that
Code:
P(X(u) = t | X(t) = u) = (n choose k) (u/t)^k (1-(u/t))^(n-k)
I have started by using the law of total probability in that P(A|B) = P(A)/P(B) although I'm not entirely convinced by this as due to the time intervals, I don't think independence can be applied.

X(t) is a poisson process with rate lambda
 

Dason

Ambassador to the humans
#2
I think you made a mistake - I doubt you're looking for P(X(u) = t | X(t) = u). Note, however, that regardless of dependence issues we can always write P(A|B) = P(A and B)/P(B)