Some notes on the Poisson distribution
Ernie Croot
October 7, 2010
1
Introduction
The Poisson distribution is one of the most important that we will encounter
in this course – it is right up there with the normal distribution. Yet, because
of time limitations, and due to the fact that its true applications are quite
technical for a course such as ours, we will not have the time to discuss them.
The purpose of this note is not to address these applications, but rather to
provide source material for some things I presented in my lectures.
2
Basic properties
Recall that
X
is a Poisson random variable with parameter
λ
if it takes on
the values 0
,
1
,
2
, ...
according to the probability distribution
p
(
x
) =
P
(
X
=
x
) =
e

λ
λ
x
x
!
.
By convention, 0! = 1.
Let us verify that this is indeed a legal probability density function (or
“mass function” as your book likes to say) by showing that the sum of
p
(
n
)
over all
n
≥
0, is 1. We have
∞
summationdisplay
n
=0
p
(
n
) =
e

λ
∞
summationdisplay
n
=0
λ
n
n
!
.
This last sum clearly is the power series formula for
e
λ
, so
∞
summationdisplay
n
=0
p
(
n
) =
e

λ
e
λ
= 1
,
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
as claimed.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Statistics, Normal Distribution, Poisson Distribution, Probability, Probability theory, probability density function

Click to edit the document details