Transcript Sullivan Chapter 6 - Whitehall District Schools

Chapter 6
Section 3
The Poisson
Probability Distribution
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 1 of 17
Poisson Distribution
● Learning objectives
1

Understand when a probability experiment follows a
Poisson process
2 Compute probabilities of a Poisson random variable
3
 Find the mean and standard deviation of a Poisson
random variable
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 2 of 17
Poisson Distribution
● Learning objectives
1

Understand when a probability experiment follows a
Poisson process
2 Compute probabilities of a Poisson random variable
3
 Find the mean and standard deviation of a Poisson
random variable
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 3 of 17
Poisson Process – Definitions
● A Poisson process: situation with the following
characteristics
 A sequence of successes, also called arrivals, appear
in time
 The probability of two or more successes in any
sufficiently small time interval is very close to zero
 The probability of success is the same for any two
time intervals of the same length
 The number of successes in any two time intervals of
the same length are independent
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 4 of 17
Poisson Process – Definitions
● A Poisson experiment has the following structure
 Customers arrive at a fixed rate
 Customers always arrive one by one
 The number of new customers arriving soon is
independent of the current situation
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 5 of 17
Poisson Process – Definitions
Example
 A ticket agent knows that customers come to the ticket office at
the rate of 5 per minute during the time period 2pm to 3pm
 Customers arrive one by one, never in groups
 Customers arrive at random, so the time until the next customer
arrives is independent of the situation now
 We count the number of customers arriving between 2:30 and
2:45
● Does this fit Poisson qualifications?
 Customers arrive at a fixed rate
 Customers always arrive one by one
 The number of new customers arriving soon is independent of
the current situation
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 6 of 17
Poisson Process – Definitions
Notation used for Poisson processes
● For a Poisson process, there is a constant rate
of successes
 This rate is called λ
 λ successes per time interval of length 1
● The random variable X counts the number of
successes in a time interval of length t
● There is no theoretical limit to the number of
successes
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 7 of 17
Poisson Process – Definitions
● In our ticket agent example
 The constant rate λ is 5 per minute
 The time interval t is 15 minutes
 X, the number of successes, is at least 0 and does
not have any theoretical upper limit
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 8 of 17
Poisson Distribution
● Learning objectives
1

Understand when a probability experiment follows a
Poisson process
2 Compute probabilities of a Poisson random variable
3
 Find the mean and standard deviation of a Poisson
random variable
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 9 of 17
Poisson Process – Calculations
● The probability distribution for the random
variable X, where
 The success rate is λ
 The time interval is t
is
( t ) x  t
P( x ) 
e
x!
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 10 of 17
Poisson Process – Calculations
● In our ticket agent example, what is the
probability of having exactly 17 customers
arriving in 3 minutes?
 λ=5
 t=3
 x = 17
( t ) x  t
P( x ) 
e
x!
(5  3)17 53 1517 15
P (17) 
e

e
 .0847
17!
17!
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 11 of 17
Poisson Process – Calculations
● If we wanted to compute the probability of
having 17 or more customers arrive in 15
minutes, we could compute
P(17) + P(18) + P(19) + …
or we could use the Complement Rule
1 – P(0) – P(1) – … – P(16)
Stop
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 12 of 17
Poisson Distribution
● Learning objectives
1

Understand when a probability experiment follows a
Poisson process
2 Compute probabilities of a Poisson random variable
3
 Find the mean and standard deviation of a Poisson
random variable
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 13 of 17
Poisson Process – Calculations
● The random variable with the probability
distribution
( t ) x  t
P( x ) 
e
x!
has a mean of
μ = λt
and a variance and standard deviation of
σ2 = λt
σ = √λt
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 14 of 17
Poisson Distribution – Mean
● We replace λt with a single parameter μ
● A Poisson random variable with mean μ is a
random variable that has the distribution
P( x ) 
x
x!
e
● This random variable has
 mean μ
 standard deviation √ μ , and
 variance μ
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 15 of 17
Poisson Distribution – Mean
Example
● If X is a Poisson random variable with mean 5,
what is the probability that X is equal to 0?
50  5
P ( 0) 
e  e 5  .0067
0!
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 16 of 17
Poisson Distribution – Summary
● A Poisson process models the number of
successes in a time interval, assuming that the
arrivals of the successes are random
● The probability of two or more successes in a
very small time interval is 0
● The probability of success in any two time
intervals of equal lengths is the same
● A Poisson random variable has mean μ and
standard deviation √ μ
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 3 – Slide 17 of 17