Transcript Sect. 1.5: Probability Distribution for Large N

The Poisson Distribution
(Not in Reif’s book)
The Poisson Probability Distribution
• "Researches on the probability of
criminal civil verdicts" 1837
• Looked at the form of the
binomial distribution
When the Number of
Trials is Large.
• He derived the cumulative
Poisson distribution as the
Limiting case of the
Binomial When the
Chance of Success
Tends to Zero.
Simeon Denis
Poisson
The Poisson Probability Distribution
• "Researches on the probability of
Simeon
Denis
criminal civil verdicts" 1837
Poisson
• Looked at the form of the binomial
distribution
When the Number of
Trials is Large.
• He derived the cumulative Poisson
distribution as the
Limiting case of the
Binomial When the
Chance of Success
Tends to Zero.
Simeon Denis “Fish”!
The Poisson Distribution
• Poisson Distribution: An approximation to the binomial
distribution for the SPECIAL CASE when the average
number (mean µ) of successes is very much smaller than
the possible number n. i.e. µ << n because p << 1.
• This distribution is important for the study of such phenomena as
radioactive decay. This distribution is NOT necessarily symmetric!
Data are usually bounded on one side & not the other.
An advantage of this distribution is that σ2 = μ
µ = 1.67
σ = 1.29
µ = 10.0
σ = 3.16
The Poisson Distribution Models Counts.
• If events happen at a constant rate over time, the Poisson
Distribution gives
The Probability of X Number of
Events Occurring in a time T.
• This distribution tells us the
Probability of All Possible Numbers of
Counts, from 0 to Infinity.
• If X= # of counts per second, then the Poisson probability that
X = k (a particular count) is:
p( X  k ) 
l
k e  
k!
λ ≡ the average number of counts per second.
Mean and Variance for the
Poisson Distribution
• It’s easy to show that for this distribution,
The Mean is:
 
• Also, it’s easy to show that The Variance is:
l
L
 
2
 The Standard Deviation is:   
 For a Poisson Distribution, the
variance and mean are equal!
More on the Poisson Distribution
Terminology: A “Poisson Process”
• The Poisson parameter  can be given as the mean
number of events that occur in a defined time period OR,
equivalently,  can be given as a rate, such as  = 2
events per month.  must often be multiplied by a time t
in a physical process
(called a “Poisson Process” )
 t
(t ) e
P( X  k ) 
k!
μ = t
σ = t
k
Example
1. If calls to your cell phone are a Poisson process
with a constant rate  = 2 calls per hour, what
is the probability that, if you forget to turn your
phone off in a 1.5 hour class, your phone rings
during that time?
Answer: If X = # calls in 1.5 hours, we want
P(X ≥ 1) = 1 – P(X = 0)
( 2 * 1.5) 0 e 2 (1.5) (3) 0 e 3
P( X  0) 
 e 3  .05
0!
0!
p
P(X ≥ 1) = 1 – .05 = 95% chance
Example Continued
2. How many phone calls do you expect
to get during the class?
<X> = t = 2(1.5) = 3
Editorial comment:
The students & the instructor in the
class will not be very
happy with you!!
Conditions Required
for the
Poisson Distribution to hold:
l
1. The rate is a constant, independent of time.
2. Two events never occur at exactly the same
time.
3. Each event is independent. That is, the
occurrence of one event does not make the
next event more or less likely to happen.
10
Example
• A production line produces 600 parts per hour with
an average of 5 defective parts an hour. If you test
every part that comes off the line in 15 minutes,what
is the probability of finding no defective parts (and
incorrectly concluding that your process is perfect)?
λ = (5 defects/hour)*(0.25 hour) =
λ = 1.25
 p(x) = (xe-)/(x!)
x = given number of defects
P(x = 0) = (1.25)0e-1.25)/(0!)
= e-1.25 = 0.287
= 28.7%
Comparison of the Binomial & Poisson
Distributions with Mean μ = 1
0.4
0.5
0.35
poisson 1
binomial N=3, p=1/3
0.3
0.2
0.3
Probability
Probability
0.4
binomial N=10,p =0.1
poiss on 1
0.25
0.2
0.15
0.1
0.1
0.05
0
0
0
1
2 m
N
3
4
5
0.0
1.0
2.0
3.0
m
4.0
5.0
6.0
7.0
N
Clearly, there is not much difference between them!
For N Large & m Fixed:
Binomial  Poisson
Poisson Distribution: As λ (Average # Counts)
gets large, this also approaches a Gaussian
λ=5
λ = 15
l
λ = 25
λ = 35