5-4 Geometric and Poisson
Distribution
Geometric Distribution
We’ve talked about these already. In a
geometric distribution the goal is
looking for first success on a particular
trial (rather than the number of
successes on a particular trial run)
Geometric Distribution
We’ve talked about these already. In a
geometric distribution the goal is
looking for first success on a particular
trial (rather than the number of
successes on a particular trial run)
P(X  x)  q
μ
1
p
and σ 
p(1-p)x-1
x 1
p
q
p
2

1 p
p
Poisson Distribution
This type of distribution is slightly different.
In a Poisson Distribution the probability of
success gets smaller and smaller as the
number of trials gets larger and larger.
Again, there are two outcomes (success or
failure) and the events must be
independent.
These are events that happen over time,
volume, area, etc… (like arrival times of
babies, airplanes, etc)
That is, waiting times.
Poisson Distribution
If λ (lambda) is the mean number of
successes over whatever interval and r
is the number of successes being
estimated, then
Poisson Distribution
If λ (lambda) is the mean number of
successes over whatever interval and r
is the number of successes being
estimated, then
P(r) 
e
λ
λ
r!
r
Poisson Distribution
If λ (lambda) is the mean number of
successes over whatever interval and r
is the number of successes being
estimated, then
With
P(r) 
e
m ean=λ
σ 
λ
λ
λ
r!
r
Calculator
Distr: poissonpdf(λ,r) = the probability of
r successes
Distr: poissoncdf(λ,r) = the probability at
least r successes
How to tell a Poisson Distribution
In our homework it will say “Explain why
a Poisson Distribution would be a good
choice for the probability distribution
of r”
Otherwise, this will be a little seen topic.
Notes
This chapter is based on Success/Failure
situations ONLY
These have all been Binomial Distributions
The Poisson Distribution can be a very
good approximation for a Probability
Distribution, provided n ≥ 100 and
np < 10.
Example
At Burnt mesa Pueblo, in one of the archaeological
excavation sites, the artifact density was 1.5
(artifacts per 10 liters of sediment). Suppose you are
going to dig up and examine 50 liters of sediment.
Let r = 0, 1, 2, 3… be a random variable that
represents the number of prehistoric artifacts found
in your 50 liters of sediment.
A) What is λ? Write out the formula for the probability distribution
of r.
B) Compute the probability that in your 50 liters of sediment you
will find 2 artifacts; 3 artifacts; 4 artifacts
C) Find the probability that you will find 3 or more artifacts in the
40 liters.
D) Find the probability that you will fewer than 3 artifacts.