Poisson distribution

ABSTRACT:

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson. The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.

INTRODUCTION

Poisson Distribution Definition:

The Poisson distribution is a discrete probability function that means the variable can only take specific values in a given list of numbers, probably infinite. A Poisson distribution measures how many times an event is likely to occur within an “x” period of time. In other words, we can define it as the probability distribution that results from the Poisson experiment. A Poisson experiment is a statistical experiment that classifies the experiment into two categories, such as success or failure. Poisson distribution is a limiting process of binomial distribution.

A Poisson random variable “x” defines the number of successes in the experiment. This distribution occurs when

there are events that do not occur as the outcomes of a definite number of outcomes. Poisson distribution is

used under certain conditions. They are:

1)The number of trials "n" tends to infinity

2)Probability of success "p" tends to zero

3)np = 1 is finite

FORMULA:

P(X−x)=(e^(−μ).μ^x)/x!

Where :

P(X−x) = Probability of x successes.

x=0,1,2,3,…

e=2.71828

μ= mean number of successes in the given time interval or region of space.

The mean and the variance of the Poisson distribution are both equal to μ.

REAL-LIFE APPLICATION OF POISSON DISTRIBUTION:

1)NUMBER OF NETWORK FAILURES PER WEEK:

Technology companies use the Poisson distribution to model the number of expected

network failures per week.

For example, suppose a given company experiences an average of 1 network

failure per week. We can use the Poisson distribution calculator to find the probability

that the company experiences a certain number of network failures in a given week:

P(X = 0 failures) = 0.36788

P(X = 1 failure) = 0.36788

P(X = 2 failures) = 0.18394

And so on.

This gives the company an idea of how many failures are likely to occur each week.

2)NUMBER OF WEBSITE VISITORS PER HOUR:

Website hosting companies use the Poisson distribution to model the number of

expected visitors per hour that websites will receive.

For example, suppose a given website receives an average of 20 visitors per

hour. We can use the Poisson distribution calculator to find the probability that

the website receives more than a certain number of visitors in a given hour:

P(X > 25 visitors) = 0.11218

P(X > 30 visitors) = 0.01347

P(X > 35 visitors) = 0.00080

And so on.

This gives hosting companies an idea of how much bandwidth to provide to different websites to ensure that they’ll be able to handle a certain number of visitors each hour.

PROBLEMS AND SOLUTIONS RELATED TO POISSON DISTRIBUTION:

A)The average number of accidents on a national highway daily is 1.8.

Determine the probability that the number of accidents

1) At atleast one

2) atmost one

Solution:

Given the average number of accidents = 1.8 = lambda value.

If we apply binomial distribution to this example, we need n and p values.

n is the number of cars going on the highway. Identifying n is not possible.

p is the probability of a car doing an accident, and p is also not possible.

P(atleast 1) = P(X >=1) = 1- P(X = 0)

P(X = 0) = (e^-1.8 (1.8)^0)/0!

= 1- 0.1653

= 0.834

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

= 0.4628

B)A distributor of bean seeds determines from the extensive test that 5 percent of a large batch of seeds will not germinate.

He sells the seeds in a package of 200 and guarantees 90 percent germination.

Determine the probability of particular packet violet the guarantee.

Solution:

Probability of seeds not germinating = 0.05 = 5 percent.

λ = mean of seeds not germinating in a sample of 200.

λ = np = 200*0.05 = 10 seeds.

given 90 percent assurance.

In a packet, if more than 20 seeds fail, we lose the assurance.

P(X > 20) = 1-P(X<= 20)

= 1 – x= 0 to 20 Σ (e^-λ λ^x)/x!

= 1 – 0.9984

= 0.0016 very fewer chances

CONCLUSION:

One particular use of the Poisson distribution is in queuing theory. In essence, the

Poisson distribution can be used to model customers arriving in a queue, such as

when checking out items at a store. It can be determined using the distribution which is

the most efficient way of organizing the queue.

Reference:

1)https://www.toppr.com/guides/maths-formulas/poisson-distribution-formula/

2)https://learningmonkey.in/courses/probability-and-statistics/lessons/poisson-distribution-real-life-examples/