Poisson Processes
Arthur White∗
14th December 2015
Poisson Distribution
Recall the Poisson distribution, whereby a discrete random variable X may take values
x = 0, 1, 2, . . . with probability
P(X = x) = P(x) =
µx −µ
e .
x!
The parameter µ, a positive real number, is called the mean of the distribution. Typically
we interpret X to be the number of events which occur during a fixed time interval, and µ
is the rate at which events are expected to occur.
Poisson Process
The Poisson process generalizes the Poisson distribution to calculate the probability of events
occurring during any arbitrary time interval. Then µ = λt, where λ is the rate per unit time
and the probability of x events is given to be
P(X = x) = P(x) =
(λt)x −λt
e .
x!
Inter-arrival Times
As well as considering the probability that a number of events will occur in a given interval,
we can also consider the converse question, the amount of time which occurs between events.
Let T1 be the time from 0 until the first event occurs. Then
P(T1 > t) = P(X = 0 in [0, t]) = e−λt .
Then
P(T1 < t) = 1 − e−λt = F1 (t).
Since Poisson events are independent, in general the time T between events is governed by
the following distribution:
∗
Based extensively on material previously taught by Eamonn Mullins.
1
CDF: F (t) = P(T < t) = 1 − e−λt
d
F (t) = λe−λt .
PDF: f (t) =
dt
These definitions are for positive time, t ≥ 0. Elsewhere, (i.e., t < 0), Fn (t) = fn (t) = 0. We
call this distribution the exponential distribution.
Properties of the Exponential Distribution
We first establish that f (t) is in fact a pdf:
Z ∞
Z ∞
f (t)dt =
λe−λt dt
0
0
Z ∞
= λ
e−λt dt
0 −λt ∞
−e
= λ
λ
0
0
= − 0−e
= e0 = 1.
Expectation of the Exponential Distribution
Next we consider
of T. Its full calculation requires using the integration by
R the expectation
R
parts formula ( udv = uv − vdu), and is left as an exercise.
Z ∞
Z ∞
1
E[T ] =
tf (t)dt =
λte−λt dt = .
λ
0
0
We sometimes refer to µ = 1/λ as a scale parameter, and can write the exponential distrit
bution in the equivalent form f (t) = µ1 e− µ .
It can also be shown that the variance of the exponential distribution is
Var[T ] =
1
.
λ2
We omit its derivation here, except to note that the calculation of E[T 2 ] requires integrating
by parts twice.
Memoryless Property
Suppose a random variable T has an exponential distribution with parameter λ, and that
after an interval of time s, no event has yet occurred. What is the probability that the event
takes a further t units of time to occur?
2
P(T > t + s|T > s) =
=
=
=
=
P(T > t + s, T > s)
P(T > s)
P(T > t + s)
P(T > s)
e−λ(t+s)
e−λs
−λt
e
P(T > t).
Thus, in general, if T follows an exponential distribution, then P(T > t + s|T > s) =
P(T > t). In words, this means that the probability that it will take at least another t units
of time for an event to occur does not depend on the length of time in which no event has
yet occurred1 . This is referred to as the memoryless property of the distribution.
1
As an analogy, imagine that you are waiting for a bus, on a cold, rainy day. While it is tempting to
think that the longer the bus takes to arrive, the more inevitable its sudden appearance will be, if the time
until the bus arrives is exponentially distributed then this will not be the case.
3