The Pure Birth Process Derivation of the Poisson Probability Distribution Assumptions events occur completely at random the probability of an event occurring is proportional to the length of time  t, say  t – is the rate of occurrence of events the probability of more than one event occurring in time  t is negligible (  0)

Derive a system of equations Let N(t) = the number of events at time t. Can move from state N(t) = n to states N(t +  t) = n, n+1 Let P n (t) = Probability of n events at time t Define the state as the number of events, n:

Let’s solve those equations

Now go to the limit…

Look, variables separable

Now look, 1 st order, linear in P 1 (t)

Look some more…

Look no further…

The Poisson Process X = a discrete random variable, the number of random occurrences (events) in time t. X = 0, 1, 2, … This was a terrific exercise. It combined differential equations with algebra and probability theory.