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Summary: The Poisson Process
A Poisson Process is a counting process, i.e. it counts
events in intervals of the form [0, t].
(First) formal definition, with 0 = t0 < t1 < t2 · · · :
A Poisson process of rate is a counting process {N(t)}t0
with:
(1) N(0) = 0
(2) independent increments: N(t1)-N(t0), N(t2)-N(t1),
etc. are independent
(3) N(s + t) - N(s) Poisson(t).
Independent of s: stationary increments.
Take s = 0 in (3):
P(N(t) = k) = e-t(t)k
k!
Also, EN(t) = t, Var N(t) = t
Interpretation of N(t):
# events that happened in time interval (0, t]
(or e.g. # defects along a stretch of cable (0, t])
1
Example 1
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