Improved Support of Random Processes in Probability 

Find the probability of Poisson process slice assuming a particular value, given process slice values at an earlier and a later time.

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X assumps = Element[{k1, k, n}, Integers] && t1 < t2 < t3 && k1 >= 0 && 0 <= k <= n;

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X pdf = Probability[ x[t2] == k1 + k \[Conditioned] {x[t1] == k1, x[t3] == k1 + n}, x \[Distributed] PoissonProcess[\[Mu]], Assumptions -> assumps]

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X FullSimplify[pdf, assumps]

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Compare the probability to PDF of binomial distribution.

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X PDF[BinomialDistribution[n, (t2 - t1)/(t3 - t1)], k]

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X PDF[BinomialDistribution[n, (t2 - t1)/(t3 - t1)], k]; FullSimplify[% == pdf, assumps]

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Find the probability of binomial process slice assuming a particular value, given process slice value at a later time.

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In[6]:=

X assumps2 = 0 <= k <= n && 0 <= t1 < t2 && n <= t2 && k <= t1;

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X pdf2 = Probability[x[t1] == k \[Conditioned] x[t2] == n, x \[Distributed] BinomialProcess[p], Assumptions -> assumps2] // FullSimplify

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Compare the probability to PDF of hypergeometric distribution.

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X PDF[HypergeometricDistribution[n, t1, t2], k]

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X PDF[HypergeometricDistribution[n, t1, t2], k]; FullSimplify[% == pdf2, assumps2]

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