The Poisson process N(t) counts the number of arrivals, up to time t, of phone calls, cosmic rays, and other discrete, unrelated events. The waiting times between the nth and (n+1)st events are independent random variables X_n with a common exponential distribution f(t) = a exp(-at). The parameter "a" is the expected number of events per unit time; that is, E(N(t)/t)=a.

A Poisson Process with E(N(t)/t)=1

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N(t)
t

Note that most of the time is spent waiting for an arrival between clusters of several events.