The Poisson Distribution
The Poisson probability distribution has the form
f(x;λ) = P(X=x) =
λxe−λ
x! , x = 0,^1 ,^2 ,^3 ,... (11 .69)
with parameter λ > 0. Here xis an integer which can increase without bound. The
Poisson probability distribution has the following properties,
1.
∑∞
x=0
λxe−λ
x! =e
−λ
(
1 + λ+
λ^2
2! +
λ^3
3! +···
)
=e−λeλ= 1
2. mean= μ=λ
3. variance σ^2 =λ
The cumulative probability function is given by
F(x;λ) =
∑x
k=0
f(k;λ) (11 .70)
The Poisson distribution occurs in application areas which record isolated events
over a period of time. For example, the number of cars entering an intersection in
a ten minute interval, the number of telephone lines in use during different periods
of the day, the number of customers waiting in line, the life expectancy of a light
bulb, the number of transistors that fail in one year, etc.
The figure 11-11 illustrates the Poisson distribution for the parameter values
λ= 1 / 2 , 1 , 2 and 3.
Figure 11-11. Selected Poisson distributions for λ=^12 , 1 , 2 , 3.
In general, the Poisson distribution is a discrete probability distribution used to
determine the number of events occurring in a fixed interval of time.
