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Compute the Probability for an Event
Compute the probability that X+Y<10 with X and Y Poisson distributed with means 1 and 2.
In[1]:=

Click for copyable input
Sum[Boole[x + y < 10] PDF[

   PoissonDistribution[Subscript[\[Lambda], 1]], x] PDF[

   PoissonDistribution[Subscript[\[Lambda], 2]], y], {x, 

  0, \[Infinity]}, {y, 0, \[Infinity]}, 

 Assumptions -> 

  Subscript[\[Lambda], 1] > 0 && Subscript[\[Lambda], 2] > 0]
Out[1]=