# 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.

 In[2]:= Xpdf = Probability[ x[t2] == k1 + k \[Conditioned] {x[t1] == k1, x[t3] == k1 + n}, x \[Distributed] PoissonProcess[\[Mu]], Assumptions -> assumps]
 Out[2]=
 In[3]:= XFullSimplify[pdf, assumps]
 Out[3]=

Compare the probability to PDF of binomial distribution.

 In[4]:= XPDF[BinomialDistribution[n, (t2 - t1)/(t3 - t1)], k]
 Out[4]=
 In[5]:= XPDF[BinomialDistribution[n, (t2 - t1)/(t3 - t1)], k]; FullSimplify[% == pdf, assumps]
 Out[5]=

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

 In[7]:= Xpdf2 = Probability[x[t1] == k \[Conditioned] x[t2] == n, x \[Distributed] BinomialProcess[p], Assumptions -> assumps2] // FullSimplify
 Out[7]=

Compare the probability to PDF of hypergeometric distribution.

 In[8]:= XPDF[HypergeometricDistribution[n, t1, t2], k]
 Out[8]=
 In[9]:= XPDF[HypergeometricDistribution[n, t1, t2], k]; FullSimplify[% == pdf2, assumps2]
 Out[9]=

## Mathematica

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