Probabilistic and Statistical Techniques 1 Lecture 19 Eng. Ismail Zakaria El Daour 2010

2 Chapter 4 (part 4) Probability Distribution Probabilistic and Statistical Techniques

3 Poisson Distributions Probabilistic and Statistical Techniques

4 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

5 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

6 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.

7 Car accidents. Number of typing errors on a page. Failure of a machine in one month. Examples Probabilistic and Statistical Techniques

Binomial distribution Mean, Variance & Standard deviation 8 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) = σ 2 = μ

9 Probabilistic and Statistical Techniques Methods for Finding Probabilities Using the Poisson Probability Formula

10 Example 1 Probabilistic 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.

11 Example 2 Probabilistic 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.

12 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.

13 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

14 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

15 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.

16 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

17 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

18 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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