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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?

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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.

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

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