Poisson Distribution Problems

Introduction to poisson distribution problems:

The Poisson Distribution is defined as a discrete distribution. It is frequently used as a model for the number of events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period. It is also using in ecological studies, e.g., to model the number of a particular family dogs found in a square mile of prairie.

Genaral Form of Poisson Distribution:

general formula for poisson distribution is

f ( n, ?) = ? n e -? ) / (n!)

where ‘e’ is the base of the natural logarithm

(e = 2.71828)

‘N’ is the number of occurrence of an event-the probability of which is given by the function

? is a positive real number

Examples:

Practice Problem 1)

On an average Friday, a waiter gets no tip from 5 customers. Find the probability that he will get no tip from 7 customers this Friday.

The waiter averages 5 customers that leave no tip on Fridays: ? = 5.

Random Variable : The number of customers that leave him no tip this Friday.so P(X = 7).

Practice Problem 2)

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

A coach can expect 3.2 injuries : ? = 3.2.

Random Variable : The counts of injuries the team has in this game.so P(X = 7).

Additional Problems : Poisson’s Distribution

Additional examples :

Practice Problem 3)

A tiny life insurance company has declared that on the average it gets 6 death claims per day. solve the probability that the insurance company gets at least seven death claims on a randomly selected day.

Answer:

P(x = 7) = 1 – P(x = 6) = 0.393697

Practice Problem 4)

The counts 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 lesser than two accidents will occur on this stretch of road during a randomly selected month.

Answer:

P(x < 2) = P(x = 0) + P(x = 1) = 0.000860

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Practice problems:

1) The amount of bridge construction projects that take place at any one time in a particular city follows a Poisson distribution with a mean of 3. calculate the probability that exactly five bridge construction projects are currently taking place in this city. (0.100819)

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