Probabilistic and Probabilistic and Statistical TechniquesStatistical Techniques

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Lecture 19

Eng. Ismail Zakaria El Daour

2010

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Chapter 4 (part 4) Probability Distribution

Probabilistic and Statistical Techniques

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Poisson Distributions

Probabilistic and Statistical Techniques

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Definition

The Poisson distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variable x is the number of occurrence of the event in an interval. The interval can be time, distance , area, volume .

Probabilistic and Statistical Techniques

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Definitions

Poisson distribution results from a procedure that meets all the following requirements:

1. The random variable x is the number of occurrence of an event over some interval.

2. The occurrence must be random.

3. The occurrence must be independent of each other.4. The occurrence must be uniformly distributed over the interval being used .

Probabilistic and Statistical Techniques

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Probabilistic and Statistical Techniques

Poisson distribution differs from a binomial distribution in these fundamental ways :

1- The binomial distribution is affected by the sample size n and the probability p . Where the Poisson distribution is affected only by the mean μ.

2- In a binomial distribution . The possible value of the random variable x are 0,1,2……n. But a Poisson distribution has possible x values of 0 ,1,2…. With no upper limit.

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• Car accidents.

•Number of typing errors on a page.

•Failure of a machine in one month.

Examples

Probabilistic and Statistical Techniques

Binomial distributionBinomial distributionMean, Variance & Standard deviationMean, Variance & Standard deviation

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Probabilistic and Statistical Techniques

If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ

Note: In a Poisson distribution, only one parameter, μ is needed to determine the probability of an event.

E(X) = μ

V(X) = σ2 = μ

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Probabilistic and Statistical Techniques

Methods for Finding ProbabilitiesUsing the Poisson Probability Formula

,...2,1,0x

!)(

e

xP 71828.2e

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Example 1Probabilistic and Statistical Techniques

The number of traffic accidents that occurs on a particular stretch of road during a month follows a Poisson distribution with a mean of 9.4. Find the probability that less than two accidents will occur on this stretch of road during a randomly selected month.

!)(

e

xP

000860.0)1()0( PP

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Example 2Probabilistic and Statistical Techniques

During a typical football game, a coach can expect 3.2 injuries. Find the probability that the team will have at most 1 injury in this game.

!)(

e

xP

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Example 3

Probabilistic and Statistical Techniques

Vehicles pass through a junction on a busy road at an average rate of 300 per hour.

• Find the probability that none passes in a given minute.• What is the expected number passing in two minutes?• Find the probability that this expected number actually pass through in a given two-minute period.

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Probabilistic and Statistical Techniques

A small life insurance company has determined that on the average it receives 6 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.

Example 4

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Example 5

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 3. Find the probability that exactly five road construction projects are currently taking place in this city

Probabilistic and Statistical Techniques

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Probabilistic and Statistical Techniques

Example 6

The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 7. Find the probability that more than four road construction projects are currently taking place in the city.

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Example 7

Suppose the number of babies born during an 8-hour shift at a hospital's maternity wing follows a Poisson distribution with a mean of 6 an hour. Find the probability that five babies are born during a particular 1-hour period in this maternity wing

Probabilistic and Statistical Techniques

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Probabilistic and Statistical Techniques

Example 8

The university policy department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 8.8 tickets per day. Find the probability that less than six tickets are written on a randomly selected day from this distribution

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The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 7. Find the probability of observing exactly three accidents on this stretch of road next month

Example 9

Probabilistic and Statistical Techniques

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