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The two are related as follows: suppose we have binomial variable with p.m.f. b(x;n,p). Suppose we increase n while simultaneously decreasing p so that the product np remains constant at some positive value l. Then as n approaches ¥ and p approaches 0, the binomial p.m.f. b(x;n,p) approaches the Poisson p.m.f.  p(x; l )

In other words, if n is "large" while p is "small" the value of the probability that X=x via the binomial distribution, may be approximated by the value of the probability that X=x via the Poisson distribution.
 

COUNTING PROCESS

A counting process {N(t), t³0) is one where we count some "event" or "occurrence" over time, with N(t) representing the total number of these that have occurred up to time t. Note that N(t)³ 0, integer, if s<t then N(s)£ N(t), and for s<t, N(t)-N(s)=no. of events in the interval (s,t).

POISSON PROCESS

A counting process is said to be a Poisson Process with rate a (>0), if

• N(0)= 0
• The number of "events" that occur in disjoint time intervals are independent of each other, i.e., the number of events in a time interval of length Dt is independent of the number of events in any other interval of time.
• The number of "events" in any time interval of length t has a Poisson distribution with mean at. Thus

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