I saw an old post here, claiming that for a Poisson Process $X(t)$:

$P[X(t) - X(s) = 1 \mid X(t) = 4]=\frac{4(t-s) s^3}{t^4}$

Am I missing something essential about stochastic processes, probability or the Poisson process? If I would have tried to solve it, I would have said:

$P[X(t) - X(s) = 1 \mid X(t) = 4]=P[4 - X(s) = 1]=P[X(s) = 3]=\frac{e^{-s\lambda}(s\lambda)^{3}}{3!}$

What am I doing wrong in the previous line?


Using the definitions, one sees that $$P[X(t) - X(s) = 1 \mid X(t) = 4]=\frac{P[X(t) - X(s) = 1, X(t) = 4]}{P[ X(t) = 4]}=\frac{P[4 - X(s) = 1, X(t) = 4]}{P[ X(t) = 4]},$$ and this would equal your suggestion $$ P[4 - X(s) = 1]$$ only if the numerator could be rewritten as $$P[4 - X(s) = 1]\,P[ X(t) = 4],$$ that is, if $X(s)$ and $X(t)$ were independent. But $X(s)$ and $X(t)$ are not independent.

  • $\begingroup$ Thank you! An important point I should get used to pay more attention to :) $\endgroup$ – Ana M Aug 14 '14 at 10:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.