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# Stats Project : Poisson Distribution

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Tweet## Rachel Miller

on 22 May 2013#### Transcript of Stats Project : Poisson Distribution

Start with the Taylor polynomial for e^x.

e^x=1 + x + x*2 / 2! + x*3 / 3! ...

Divide both sides by e^x.

1=1/(e^x)+x/(e^x)+x*2/(e^x)...

p(k) = r*k / (k!)(e*r)

where k is the nth term/number you want to occur and r is the mean. Definition of "Poisson Distribution" Discrete probability distribution

Expresses the probability that a given # of events will occur during a fixed interval of time and/or space

Predicts the degree of spread around a known average rate of occurrence Equation Binomial distribution involves two parameters, n and p.

Binomial is used to model the occurrence of an event in n amount of trials.

Each trial results in either a success or a failure.

Each trial has the same probability of success, p

They are independent trials. What's the difference between a Binomial Distribution and a Poisson Distribution? What areas is it used in? electrical system

astronomy

biology

business management

civil engineering

finance and insurance

earthquake seismology

radioactivity Siméon Denis Poisson (1781–1840) Meet the Creator Poisson Distribution By: Flora Lam, Lawrence Leung, Benson Li, Annette Luo, Rachel Miller S'2013 Project (AP Statistics 6/7) Last name pronounced similarly to croissant

Was a French mathematician and physicist

Published his distribution in 1837

Poisson distribution has one parameter, m, which is the average # of events in a unit of measure.

It models the rates that an event has occurred. In this case: the # of occurrences in a unit of measure.

Events can happen at any point, the probability of which is very small.

The average number of events is constant over a unit of measure.

They are independent events.

Binomial distribution within a given time frame (probability of the frequency of an event during a given time span) Examples :

pieces of mail per day, phone calls per hour Simply Put :

when the event is something that can be counted in whole numbers

when occurrences are independent

when the average frequency of occurrence for the time period in question is known

when it is possible to count how many events have occurred When To Use It Derivation of Formula Sample Problem 1 A manufacturer of TV says that their product is on average 5% defective. What is the probability that of a sample of 100 TVs, only two at most will be defective?

Let X=number of success. r=np=100(.05)=5

P(X<3)=P(0)+P(1)+P(2)

1 - e^(-5) *[1 + 5 + 25/2 ] = 0.12395 Sample Problem 2 Sample Problem 2 Conditions:

Events are occurring independently in time

The probability that an event occurs in a given length of time doesn't change Problem:

Asteroids with a diameter of at least 1 km collide with the earth at a rate of approximately 2 per million years.

What is the probability that in a randomly selected million year period, there is exactly one collision? Equation: x=1

λ=2 Sample Problem 2 Equation x=1, λ=2

p(x=1)= (2^1)(e^-2)/(1!)=0.2707 Sample Problem 2 Poisson Distribution Graph for λ=2 Probability using binomial distribution: .118

Full transcripte^x=1 + x + x*2 / 2! + x*3 / 3! ...

Divide both sides by e^x.

1=1/(e^x)+x/(e^x)+x*2/(e^x)...

p(k) = r*k / (k!)(e*r)

where k is the nth term/number you want to occur and r is the mean. Definition of "Poisson Distribution" Discrete probability distribution

Expresses the probability that a given # of events will occur during a fixed interval of time and/or space

Predicts the degree of spread around a known average rate of occurrence Equation Binomial distribution involves two parameters, n and p.

Binomial is used to model the occurrence of an event in n amount of trials.

Each trial results in either a success or a failure.

Each trial has the same probability of success, p

They are independent trials. What's the difference between a Binomial Distribution and a Poisson Distribution? What areas is it used in? electrical system

astronomy

biology

business management

civil engineering

finance and insurance

earthquake seismology

radioactivity Siméon Denis Poisson (1781–1840) Meet the Creator Poisson Distribution By: Flora Lam, Lawrence Leung, Benson Li, Annette Luo, Rachel Miller S'2013 Project (AP Statistics 6/7) Last name pronounced similarly to croissant

Was a French mathematician and physicist

Published his distribution in 1837

Poisson distribution has one parameter, m, which is the average # of events in a unit of measure.

It models the rates that an event has occurred. In this case: the # of occurrences in a unit of measure.

Events can happen at any point, the probability of which is very small.

The average number of events is constant over a unit of measure.

They are independent events.

Binomial distribution within a given time frame (probability of the frequency of an event during a given time span) Examples :

pieces of mail per day, phone calls per hour Simply Put :

when the event is something that can be counted in whole numbers

when occurrences are independent

when the average frequency of occurrence for the time period in question is known

when it is possible to count how many events have occurred When To Use It Derivation of Formula Sample Problem 1 A manufacturer of TV says that their product is on average 5% defective. What is the probability that of a sample of 100 TVs, only two at most will be defective?

Let X=number of success. r=np=100(.05)=5

P(X<3)=P(0)+P(1)+P(2)

1 - e^(-5) *[1 + 5 + 25/2 ] = 0.12395 Sample Problem 2 Sample Problem 2 Conditions:

Events are occurring independently in time

The probability that an event occurs in a given length of time doesn't change Problem:

Asteroids with a diameter of at least 1 km collide with the earth at a rate of approximately 2 per million years.

What is the probability that in a randomly selected million year period, there is exactly one collision? Equation: x=1

λ=2 Sample Problem 2 Equation x=1, λ=2

p(x=1)= (2^1)(e^-2)/(1!)=0.2707 Sample Problem 2 Poisson Distribution Graph for λ=2 Probability using binomial distribution: .118