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## Poisson Distribution

Poisson distribution is a theoretical discrete probability and is also known as the Poisson distribution probability mass function. It is used to find the probability of an independent event that is occurring in a fixed interval of time and has a constant mean rate. The Poisson distribution probability mass function can also be used in other fixed intervals such as volume, area, distance, etc. A Poisson random variable will relatively describe a phenomenon if there are few successes over many trials. The Poisson distribution is used as a limiting case of the binomial distribution when the trials are large indefinitely. If a Poisson distribution models the same binomial phenomenon, λ is replaced by np. Poisson distribution is named after the French mathematician Denis Poisson.

## What is Poisson Distribution?

Poisson distribution definition is used to model a discrete probability of an event where independent events are occurring in a fixed interval of time and have a known constant mean rate. In other words, Poisson distribution is used to estimate how many times an event is likely to occur within the given period of time. λ is the Poisson rate parameter that indicates the expected value of the average number of events in the fixed time interval. Poisson distribution has wide use in the fields of business as well as in biology.

Let us try and understand this with an example, customer care center receives 100 calls per hour, 8 hours a day. As we can see that the calls are independent of each other. The probability of the number of calls per minute has a Poisson probability distribution. There can be any number of calls per minute irrespective of the number of calls received in the previous minute. Below is the curve of the probabilities for a fixed value of λ of a function following Poisson distribution:

If we are to find the probability that more than 150 calls could be received per hour, the call center could improve its standards on customer care by employing more services and catering to the needs of its customers, based on the understanding of the Poisson distribution.

## Poisson Distribution Formula

Poisson distribution formula is used to find the probability of an event that happens independently, discretely over a fixed time period, when the mean rate of occurrence is constant over time. The Poisson distribution formula is applied when there is a large number of possible outcomes. For a random discrete variable X that follows the Poisson distribution, and λ is the average rate of value, then the probability of x is given by:

f(x) = P(X=x) = (e^{-λ} λ^{x} )/x!

Where

- x = 0, 1, 2, 3…
- e is the Euler’s number(e = 2.718)
- λ is an average rate of the expected value and λ = variance, also λ>0

### Poisson Distribution Mean and Variance

For Poisson distribution, which has λ as the average rate, for a fixed interval of time, then the mean of the Poisson distribution and the value of variance will be the same. So for X following Poisson distribution, we can say that λ is the mean as well as the variance of the distribution.

Hence: E(X) = V(X) = λ

where

- E(X) is the expected mean
- V(X) is the variance
- λ > 0

## Properties of Poisson Distribution

The Poisson distribution is applicable in events that have a large number of rare and independent possible events. The following are the properties of the Poisson Distribution. In the Poisson distribution,

- The events are independent.
- The average number of successes in the given period of time alone can occur. No two events can occur at the same time.
- The Poisson distribution is limited when the number of trials n is indefinitely large.
- mean = variance = λ
- np = λ is finite, where λ is constant.
- The standard deviation is always equal to the square root of the mean μ.
- The exact probability that the random variable X with mean μ =a is given by P(X= a) = μ
^{a }/ a! e^{-μ} - If the mean is large, then the Poisson distribution is approximately a normal distribution.

## Poisson Distribution Table

Similar to the binomial distribution, we can have a Poisson distribution table which will help us to quickly find the probability mass function of an event that follows the Poisson distribution. The Poisson distribution table shows different values of Poisson distribution for various values of λ, where λ>0. Here in the table given below, we can see that, for P(X =0) and λ = 0.5, the value of the probability mass function is 0.6065 or 60.65%.

## Applications of Poisson Distribution

There are various applications of the Poisson distribution. The random variables that follow a Poisson distribution are as follows:

- To count the number of defects of a finished product
- To count the number of deaths in a country by any disease or natural calamity
- To count the number of infected plants in the field
- To count the number of bacteria in the organisms or the radioactive decay in atoms
- To calculate the waiting time between the events.

**Important Notes**

- The formula for Poisson distribution is f(x) = P(X=x) = (e
^{-λ}λ^{x})/x!. - For the Poisson distribution, λ is always greater than 0.
- For Poisson distribution, the mean and the variance of the distribution are equal.

## Poisson Distribution Examples

**Example 1: **In a cafe, the customer arrives at a mean rate of 2 per min. Find the probability of arrival of 5 customers in 1 minute using the Poisson distribution formula.

**Solution:**

Given: λ = 2, and x = 5.

Using the Poisson distribution formula:

P(X = x) = (e^{-λ} λ^{x} )/x!

P(X = 5) = (e^{-2} 2^{5} )/5!

P(X = 6) = 0.036

**Answer: The probability of arrival of 5 customers per minute is 3.6%.**

**Example 2: **Find the mass probability of function at x = 6, if the value of the mean is 3.4.

**Solution**:

Given: λ = 3.4, and x = 6.

Using the Poisson distribution formula:

P(X = x) = (e^{-λ} λ^{x} )/x!

P(X = 6) = (e^{-3.4} 3.4^{6} )/6!

P(X = 6) = 0.072

**Answer: The probability of function is 7.2%.**

**Example 3: If 3% of electronic units manufactured by a company are defective. Find the probability that in a sample of 200 units, less than 2 bulbs are defective. **

**Solution:**

The probability of defective units p = 3/100 = 0.03

Give n = 200.

We observe that p is small and n is large here. Thus it is a Poisson distribution.

Mean λ= np = 200 × 0.03 = 6

P(X= x) is given by the Poisson Distribution Formula as (e^{-λ} λ^{x} )/x!

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

=(e^{-6} 6^{0} )/0! + (e^{–6}6^{1} )/1!

= e^{-6} + e^{–6 }× 6

= 0.00247 + 0.0148

P(X < 2) = 0.01727

**Answer: The probability that less than 2 bulbs are defective is 0.01727**

## FAQs on Poisson Distribution

### What is Poisson Distribution?

Poisson distribution definition says that it is a discrete probability of an event where independent events are occurring in a fixed interval of time and has a known constant mean rate. In other words, for a fixed interval of time, a Poisson distribution can be used to measure the probability of the occurrence of an event. Poisson distribution has wide use in the field of business as well in biology.

### What is Lambda in Poisson Distribution?

In Poisson distribution, lambda is the average rate of value for a function. It is also known as the mean of the Poisson distribution. For Poisson distribution, variance is also the same as the mean of the function hence lambda is also the variance of the function that follows the Poisson distribution.

### What are the Characteristics of Poisson Distribution?

The basic characteristic of a Poisson distribution is that it is a discrete probability of an event. Events in the Poisson distribution are independent. The occurrence of the events is defined for a fixed interval of time. The value of lambda is always greater than 0 for the Poisson distribution.

### What is the Difference Between Poisson Distribution and Binomial Distribution?

For Poisson distribution, the sample size is unknown but for the binomial distribution, the sample size is fixed. Poisson distribution can have any value in the sample size and is always greater than 0, whereas Binomial distribution has a fixed set of values in the sample size.

### How to Calculate Poisson Distribution?

Poisson distribution is calculated by using the Poisson distribution formula. The formula for the probability of a function following Poisson distribution is: f(x) = P(X=x) = (e^{-λ} λ^{x} )/x!

Where

- x = 0, 1, 2, 3…
- e is the Euler’s number
- λ is an average rate of value and variance, also λ>0

### Where Do We Use Poisson Distribution?

Poisson distribution is used in many fields. It has wide use in the field of business. Businessmen use it to predict the future of the business, growth, and decay of the business. Poisson distribution is used in biology especially estimating the number of offsprings in mutation after a fixed period of time.

In Statistics, Poisson distribution is one of the important topics. It is used for calculating the possibilities for an event with the average rate of value. Poisson distribution is a discrete probability distribution. In this article, we are going to discuss the definition, Poisson distribution formula, table, mean and variance, and examples in detail.

## 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 “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 the 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:

- The 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} λ^{x})/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. The table 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.

## Poisson Distribution Mean and Variance

Assume that, we conduct a Poisson experiment, in which the average number of successes within a given range is taken 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. Then, the Poisson probability is:

**P(x, λ ) =(e ^{– λ} λ^{x})/x!**

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

For a Poisson Distribution, the mean and the variance are equal. It means that **E(X) = V(X)**

Where,

V(X) is the variance.

## Poisson Distribution Expected Value

A random variable is said to have a Poisson distribution with the parameter λ, where “λ” is considered as an expected value of the Poisson distribution.

The expected value of the Poisson distribution is given as follows:

E(x) = μ = d(e^{λ(t-1)})/dt, at t=1.

E(x) = λ

Therefore, the expected value (mean) and the variance of the Poisson distribution is equal to λ.

### Poisson Distribution Examples

An example to find the probability using the Poisson distribution is given below:

**Example 1:**

A random variable X has a Poisson distribution with parameter λ such that P (X = 1) = (0.2) P (X = 2). Find P (X = 0).

**Solution:**

For the Poisson distribution, the probability function is defined as:

P (X =x) = (e^{– λ} λ^{x})/x!, where λ is a parameter.

Given that, P (x = 1) = (0.2) P (X = 2)

(e^{– λ} λ^{1})/1! = (0.2)(e^{– λ} λ^{2})/2!

⇒λ = λ^{2}/ 10

⇒λ = 10

Now, substitute λ = 10, in the formula, we get:

P (X =0 ) = (e^{– λ} λ^{0})/0!

P (X =0) = e^{-10 }= 0.0000454

Thus, P (X= 0) = 0.0000454

**Example 2**:

Telephone calls arrive at an exchange according to the Poisson process at a rate λ= 2/min. Calculate the probability that exactly two calls will be received during each of the first 5 minutes of the hour.

**Solution:**

Assume that “N” be the number of calls received during a 1 minute period.

Therefore,

P(N= 2) = (e^{-2}. 2^{2})/2!

P(N=2) = 2e^{-2}.

Now, “M” be the number of minutes among 5 minutes considered, during which exactly 2 calls will be received. Thus “M” follows a binomial distribution with parameters n=5 and p= 2e^{-2}.

P(M=5) = 32 x e^{-10}

P(M =5) = 0.00145, where “e” is a constant, which is approximately equal to 2.718.

Frequently Asked Questions on Poisson Distribution

### What is a Poisson distribution?

A Poisson distribution is defined as a discrete frequency distribution that gives the probability of the number of independent events that occur in the fixed time.

### When do we use Poisson distribution?

Poisson distribution is used when the independent events occurring at a constant rate within the given interval of time are provided.

### What is the difference between the Poisson distribution and normal distribution?

The major difference between the Poisson distribution and the normal distribution is that the Poisson distribution is discrete whereas the normal distribution is continuous. If the mean of the Poisson distribution becomes larger, then the Poisson distribution is similar to the normal distribution.

### Are the mean and variance of the Poisson distribution the same?

The mean and the variance of the Poisson distribution are the same, which is equal to the average number of successes that occur in the given interval of time.

### Mention the three important constraints in Poisson distribution.

The three important constraints used in Poisson distribution are:

The number of trials (n) tends to infinity

The probability of success (p) tends to zero

np=1, which is finite.

**Poisson Distribution Example**

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

*Solution:* This is a Poisson experiment in which we know the following:

- μ = 2; since 2 homes are sold per day, on average.
- x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.
- e = 2.71828; since
*e*is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(*x*; μ) = (e^{-μ}) (μ^{x}) / x!

P(3; 2) = (2.71828^{-2}) (2^{3}) / 3!

P(3; 2) = (0.13534) (8) / 6

P(3; 2) = 0.180

Thus, the probability of selling 3 homes tomorrow is 0.180 .

## Cumulative Poisson Probability

A **cumulative Poisson probability** refers to the probability that the Poisson random variable is greater than some specified lower limit and less than some specified upper limit.

**Cumulative Poisson Example**

Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

*Solution:* This is a Poisson experiment in which we know the following:

- μ = 5; since 5 lions are seen per safari, on average.
- x = 0, 1, 2, or 3; since we want to find the likelihood that tourists will see fewer than 4 lions; that is, we want the probability that they will see 0, 1, 2, or 3 lions.
- e = 2.71828; since
*e*is a constant equal to approximately 2.71828.

To solve this problem, we need to find the probability that tourists will see 0, 1, 2, or 3 lions. Thus, we need to calculate the sum of four probabilities: P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5). To compute this sum, we use the Poisson formula:

P(x < 3, 5) = P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5)

P(x < 3, 5) = [ (e^{-5})(5^{0}) / 0! ] + [ (e^{-5})(5^{1}) / 1! ] + [ (e^{-5})(5^{2}) / 2! ] + [ (e^{-5})(5^{3}) / 3! ]

P(x < 3, 5) = [ (0.006738)(1) / 1 ] + [ (0.006738)(5) / 1 ] + [ (0.006738)(25) / 2 ] + [ (0.006738)(125) / 6 ]

P(x < 3, 5) = [ 0.0067 ] + [ 0.03369 ] + [ 0.084224 ] + [ 0.140375 ]

P(x < 3, 5) = 0.2650

Thus, the probability of seeing at no more than 3 lions is 0.2650.

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