Lesson 6 - 3
Poisson Probability Distribution
Objectives
• Understand when a probability experiment follows a
Poisson process
• Compute probabilities of a Poisson random variable
• Find the mean and standard deviation of a Poisson
random variable
Vocabulary
• Poisson process – used to computer probabilities of
experiments in which the random variable X counts
the number of occurrences (successes) of a
particular event with in a specified interval (usually
time or space)
If we examine the binomial distribution as the
number of trials gets larger and larger while the
probability of success p gets smaller and smaller,
we observe the Poisson Distribution (Simeon Denis
Poisson). This distribution deals with the
probabilities of rare events that occur infrequently in
space, time, distance, area, volume, and so forth.
Criteria for a Poisson Probability
Experiment
An experiment is said to be a Poisson experiment
provided:
1. The probability of two or more successes in any
sufficiently small subinterval* is 0
2. The probability of success is the same for any two
intervals of equal length
3. The number of successes in any interval is
independent of the number of successes in any
other interval provided the intervals are not
overlapping
* - for example, the fixed interval might be any time
between 0 and 5 minutes. A subinterval could be
any time between 1 and 2 minutes
Poisson PDF
If X is the number of successes in an interval of fixed
length t, then probability formula for X is
(λt)x
P(x) = --------- e-λt
x = 0, 1, 2, 3, …
x!
where λ (the Greek letter lamda) represents the average
number of occurrences of the event in some interval of
length 1 and e = 2.71828.... (Euler's constant)
If X is the number of successes in an interval of fixed
length and X follows a Poisson process with mean μ, the
probability distribution function (PDF) for X is
μx
P(x) = --------- e-μ
x!
x = 0, 1, 2, 3, …
Poisson PDF (cont)
Mean (or Expected Value) and Standard Deviation :
A random variable X that follows a Poisson process with
parameter λ has a mean (or expected value) and standard
deviation given by the formulas below:
Mean: μx = λt
Standard Deviation: σx = √λt = √μx
where t is the length of the interval
Note: At least probabilities must be computed using the
Complement rule for Poisson probabilities
Calculator: 2nd VARS
poissonpdf(μ,x)
poissoncdf(μ,x)
Examples of a Poisson
a) The number of accidents that occur per month (or
week or day, etc) at a given intersection or in a
manufacturing plant.
b) The number of arrivals per minute (or hour, etc.) at a
medical clinic or other servicing facility such as a
garage, hospital, bank airport, etc. Number of cars
arriving at a toll booth per day ( or hour or minute)
c) The number of defects detected by quality control
per day(per foot, per yard of material, per batch, etc.)
Example 1
The number of accidents in an office building during a
four week period is averages 2. What is the probability
that there will be one or fewer accidents in the next fourweek period?
μ=2
μx
P(x) = --------- e-μ
x!
P(1 or fewer) = P(0) + P(1)
20
P(0) = --------- e-2 = e-2 = 0.13534
0!
21
P(1) = --------- e-2 = 2e-2 = 0.27067
1!
P(1 or fewer) = P(0) + P(1) = 0.13534 + 0.27067 = 0.40602
Example 2
Daily demand for a certain replacement part for a VCR
μx
averages 0.9.
-μ
P(x) = --------- e
x!
μ = 0.9
a) What is the probability that no replacements parts
will be needed on a particular day?
(0.9)0
P(0) = --------- e-(0.9)
0!
= e-0.9 = 0.4066
b) That no more than three will be needed?
P(no more than 3) = P(x ≤ 3) = P(0) + P(1) + P(2) + P(3)
(0.9)1
P(1) = --------- e-(0.9) = 0.3659
1!
(0.9)3
P(3) = --------- e-(0.9) = 0.0494
3!
(0.9)2
P(2) = --------- e-(0.9) = 0.1647
2!
= 0.9866
Example 3
The number of calls to a police department between
8pm and 8:30pm on Friday averages 3.5.
μx
P(x) = --------- e-μ
x!
μ = 3.5
a) What is the probability of no calls during this
period?
3.50
P(0) = --------- e-3.5
0!
= 0.0302
b) Is it likely that the police will get 7 calls?
3.57
P(7) = --------- e-3.5
7!
= 0.0386
< 5% so not likely!
c) What is the mean and standard deviation of the
number of calls?
μ = 3.5
σ = 3.5 = 1.8708
Summary and Homework
• Summary
– Poisson process has 3 criteria
– Used for occurrences over time or space
– Calculator has pdf and cdf functions
• Homework
– pg 348 - 350: 3, 6, 9, 10, 14, 17