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.
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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.
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