Poisson Distribution

What is Poisson Distribution?

Mathematics is a vast subject with different topics under it. Probability theory and combination theory are the two most prominent theories. Poisson distribution theory is a part of probability that came from the name of a French mathematician Simeon Denis Poisson. A number of events that express success or failure are the theory of Poisson distribution. The time interval in probability is a crucial thing that determines a lot of factors and the result. 

Poisson distribution theory tells us about the discrete probability distribution, which means the likelihood of an event to occur in a fixed time interval or events that occur in constant and independently of the time in relation to the last event. For number events like area, distance, and volume Poisson distribution can be used.

Poisson Distribution Examples

A person is keeping a record of the number of letters he receives each day, and he might notice that he on average is receiving four letters in a day. When accepting any piece of a letter does not affect the time of arrival of the future letter then it is assumed that the number of the letter received in a day obeys Poisson distribution. This means the arrival of one letter is independent of the other letter in the future. You can also take the example of calls from the call centre that are made in a day and all are independent of each other. This is one of the easiest poisson distribution examples to understand. 

In this article, we will discuss in detail Poisson distribution and Poisson distribution table, means and variance. Well, there are many such examples that can be drawn from your daily life, which shows the implication of Poisson distribution theory.

Let’s try to understand what is Poisson distribution and what is Poisson distribution used for? let us start with the definition

Poisson Distribution Definition

Poisson distribution is categorized into two categories success and failure. Poisson distribution is the result of the Poisson experiment, and it is a statistical experiment.  Poisson distribution definition clarifies the value of probability theory of probability. To understand the theory clearly one should have clear knowledge and understanding of statistics and binomial equations. These few things will make your understanding of the theory simple. It is a limited process of binomial distribution and occurrence of success and failure. ‘x’ in a Poisson distribution represents the number of successes in the experiment. There is a certain condition under which Poisson distribution occurs. The distribution occurs when the result of the outcome does not occur or a specific number of outcomes. These can be denoted by:

  • Number of trials “n” tends to infinity

  • Probability of success “p” tends to zero

  • np = 1 is finite

Poisson Distribution Formula

The formula for the Poisson distribution function is given by:

f(x) =(e– λ λx)/x!

Where,

e is the base of the logarithm

x is a Poisson random variable

λ is an average rate of value 

Poisson Distribution Table

As with the binomial distribution, there is a table that we can use under certain conditions that will make calculating probabilities a little easier when using the Poisson distribution. Poisson distribution table helps you to solve Poisson distribution questions.

The chart is showing the values of f(x) = P(X ≥ x), where X has a Poisson distribution with parameter λ. Refer the values from the table and substitute it in the Poisson distribution formula to get the probability value. The table displays the values of the Poisson distribution.

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The respective image shows the poisson distribution table for the better understanding of further equations. 

Poisson Distribution Mean and Variance

Let assume that we will conduct a Poisson experiment in which the average number of successes is taken as a range that is denoted as λ. In Poisson distribution, the mean of the distribution is represented by λ and e is constant, which is approximately equal to 2.71828. For the given equation, the Poisson probability will be:

P(x, λ ) =(e– λ λx)/x!

In Poisson distribution, the mean is represented as E(X) = λ.

The mean and the variance of Poisson Distribution are equal. It means that E(X) = V(X)

Where,

V(X) is the variance.

Solved Examples

Example 1

If the random variable X follows a Poisson distribution with mean 

3.4, find P(X=6)?

You can write it as: 

if the random variable X follows a Poisson distribution with mean 3,4 find P (X= 6)

Solution

This can be written more quickly as: if X - Po(3.4) find P(X=6)

Now,

p(x=6) = \[\frac{e^{-λ}λ^{6}}{6!}\]

             = \[\frac{e^{-3.4}3.4^{6}}{6!}\] (mean,λ=3.4)

= 0.07160 = 0.072 (to 3 d.p)

Example 2

The number of industrial injuries per working week in a particular factory is known to follow Poisson Distribution with mean 0.5

Find the probability that

  1. In a particular week, there will be

             Less than 2 accidents

             More than 2 accidents

  1. In a three week period, there will be no accidents

Solution

Let A be the number of accidents in one week so A- Po (0.5)

(a)  (i) P(A<2) = P(A≤1)

                        = 0.9098 (from tables in Appendix 3(p257), to 4 d.p.)

                or, from the formula,

           P(A<2) = P(A=0) + P(A=1)

                        = \[e^{-0.5}\] + \[\frac{e^{-0.5}0.5}{1!}\]

                         = \[\frac{3}{2}\] \[e^{-0.5}\]

                         = 0.9098.

     (ii ) P(A<2) = 1 -  P(A≤2)

                        = 1 - 0.9856 (from tables)

                        = 0.0144 (to 4 d.p.)

                 or        

          1-[P(A=0) + P(A=1) + P(A=2)]        

                       = 1-\[e^{-0.5}\] +  \[e^{-0.5}\]0.5 +  \[\frac{e^{-0.5}0.5}{2!}\]

                       = 1-\[e^{-0.5}\](1+0.5+0.125)

                       = 1-1.625 \[e^{-0.5}\]

                       = 0.0144.

(b) P(0 in 3 weeks) = (\[e^{-0.5}\])³ = 0.223

FAQ (Frequently Asked Questions)

1. What is the Poisson Distribution used for?

There are various tools of probability theory, and one of the tools is the Poisson theory. It predicts certain events to happen in future. The events are a result of a fixed time interval and give the probability of future success and failure. Poisson distribution works only on integers on a horizontal axis. It can be used to find out area, volume, and distance in a given event. The events are independent that occurs, and it will be time-specific. It will be distributed in a statistical representation in a graphical manner. You will find the application of Poisson distribution in business, statistics, and daily life, which makes it vital for daily use. Poisson distribution questions become easy to solve when you have your concepts clear on statistics. 

2. What are the Conditions of Poisson Distribution?

There is a certain Poisson distribution assumption that needs to satisfy for the theory to be valid. The conditions are as follows:


  • In a fixed time interval, any number of events can occur.

  • All the events should be independent of one another.

  • The rate of occurrence of each event should be a constant rate, and the rate of the events should not change with the change in time.

  • The probability of the length of the time is proportional to the occurrence of the event is a fixed period of time.