Poisson Distribution – Definition, Formula, Table and Examples

Poisson Distribution is one of the types of discrete probability distributions like binomial distribution in probability. It expresses the probability of a given number of events occurring in a fixed interval of time.

The Poisson distribution is a type of discrete probability distribution that determines the likelihood of an event occurring a specific number of times (k) within a designated time or space interval. This distribution is characterized by a single parameter, λ (lambda), representing the average number of occurrences of the event.

In this article, we will discuss the Poisson Distribution including its definition, formula, examples, and properties of Poisson Distribution in detail.

What is Poisson Distribution?

Poisson distribution is used to model the number of events that occur in a fixed interval of time or space, given the average rate of occurrence, assuming that the events happen independently and at a constant rate.

It deals with discrete random variables, meaning the number of events can only take on non-negative integer values (0, 1, 2, 3,…). Each event is considered to be independent of others and they are assumed to occur at a constant average rate (λ) over the given interval.

Poisson Distribution Definition

Poisson distribution is a mathematical concept used to model the probability of a given number of events occurring within a fixed interval of time or space, provided that these events happen at a constant average rate and are independent of the time since the last event.

Poisson Distribution Formula

The Poisson distribution is characterized by a single parameter, lambda (λ), which represents the average rate of occurrence of the events. The probability mass function of the Poisson distribution is given by:

P (X = k) = e^−λλ^k​ / k!

Where:

• P(X=k) is the probability of observing k events
• e is the base of the natural logarithm (approximately 2.71828)
• λ is the average rate of occurrence of the events
• k is the number of events that occur
• k! denotes the factorial of k

Properties of Poisson Distribution

• Probability Mass Function (PMF): PMF describes the likelihood of observing a specific number of events in a fixed interval. It is given by:
  P(X = k) = (e^-λ × λ^k) / k!

• Cumulative Distribution Function (CDF): CDF gives the probability that the random variable is less than or equal to a certain value. It is expressed as:
  F(x) = ∑(from k=0 to ⌊x⌋) (e^-λ × λ^k) / k!

• Moment Generating Function (MGF): MGF provides a way to derive moments of the distribution. It is represented by:
  M(t) = e^{(λ(e^t – 1))}

• Characteristic Function (CF): CF is an alternative way to describe the distribution and is given by:
  ϕ(t) = e^{(λ(e^(it) – 1))}

• Probability Generating Function (PGF): PGF generates the probabilities of the distribution and is expressed as:
  G(z) = e^{(λ(z – 1))}

• Median: Median, which represents the central value, is approximately λ+ (1/3)​−0.02/λ​.

• Mode: Mode, or the most probable value, is simply the integer part of λ, denoted as ⌊λ⌋.

• Mean and Variance: The mean (λ) and variance (λ) of a Poisson distribution are equal. This means that both the average number of events and the spread or variability around this average are characterized by the same parameter.

• Non-negative and Discrete: The Poisson distribution describes the probability of non-negative integer values only, as it models counts of events. It is a discrete probability distribution.

• Memorylessness: Events in a Poisson process are memoryless, meaning the probability of an event occurring in the future is independent of the past, given the current state. For example, if you’re waiting for a bus, the probability of the bus arriving in the next minute doesn’t depend on how long you’ve already been waiting.

• Independent Increments: The number of events occurring in non-overlapping intervals is independent. For instance, if you’re counting the number of cars passing through an intersection in one minute, the number of cars in the next minute is independent of the number in the previous minute.

• Rare Events Approximation: When the average rate of occurrence (λ) is large and the probability of a single event is small, the Poisson distribution can approximate the binomial distribution. This is known as the “rare events” approximation, where the binomial distribution with a large number of trials and a small probability of success converges to a Poisson distribution.

• Skewness and Kurtosis: Poisson distribution is positively skewed (skewness > 0) and leptokurtic (kurtosis > 0), meaning it has a longer tail on the right side and heavier tails than the normal distribution. However, for large values of λ, it becomes increasingly symmetric and bell-shaped, resembling a normal distribution.

Some other properties are:

• Poisson distribution has only one parameter “λ” where λ = np.
• Poisson distribution is positively skewed and leptokurtic.

Note: Here leptokurtic means values greater kurtosis than the normal distribution, and kurtosis is the nothing but the sharpness of the peak of the frequency distribution curve.

Poisson Distribution Table

A Poisson distribution table is a tabulation of probabilities for a Poisson distribution and probabilities here can be calculated using the Probability Mass Function of Poisson Distribution which is given by [Tex]\bold{\text{PMF} = \frac{\lambda^k e^{-\lambda}}{k !}}[/Tex]. The following table is one such example of the Poisson Distribution Table.

k (Number of Events)	P(X = k)
0	0.0498
1	0.1494
2	0.2241
3	0.2241
4	0.1681
5	0.1009
6	0.0505
7	0.0214
8	0.0080
9	0.0027
10	0.0008

Poisson Distribution Graph

The following illustration shows the Graph of Poisson Distribution or Poisson Distribution Curve.

Poisson Distribution Mean and Variance

In the Poisson distribution, both the mean (average) and variance are equal and are denoted by the parameter λ (lambda). This property of equal mean and variance is a distinctive characteristic of the Poisson distribution and simplifies its statistical analysis.

Poisson Distribution Mean

Mean of a Poisson distribution is also known as Poisson Distribution expected value or average of the distribution and is represented by “E[X]” or “λ” (lambda). This means that the mean for poison distribution is equal to the parameter i.e., λ. Mathematically, this equation is represented as follows:

E[X] = λ

where,

• E[X] is the mean of the Poisson’s Distribution
• λ is the parameter of the distribution
• X is a random variable following a Poisson distribution

Other than this, we have one more formula for the mean of expectation of the distribution that is:

Mean = λ = np

where,

• n is the number of trails
• p is the probability of success

Poisson Distribution Variance

Variance is the measure of the spread or dispersion of the random variable around its mean. For Poisson Distribution, variance is equal to the parameter λ (lambda).

Thus, the variance of a Poisson distribution can be expressed as:

Var(X) = λ

where,

• Var(X) is the variance of the Poisson-distributed random variable X
• λ is the parameter of the Poisson distribution

Standard Deviation of Poisson Distribution

Standard Deviation of a poisson distribution is a measure of the amount of variability or dispersion in the distribution. Mathematically, it is given by:

σ = √λ

where

• λ (lambda) is the average rate of occurrence of events
• σ (sigma) is the standard deviation of the distribution

Probability Mass Function of Poisson Distribution

The Probability Mass Function for Poisson Distribution is given by:

[Tex]\bold{\text{PMF} = \frac{\lambda^k e^{-\lambda}}{k !}}[/Tex]

where,

• λ is the parameter which is also equal to mean and variance as well
• k is the number of times an event occurs
• e is Euler’s number ( ≈2.718)

Difference between Binomial and Poisson Distribution

The key differences between Poisson Distribution and Binomial Distribution are listed in the following table:

Difference between Binomial and Poisson Distribution
Aspect	Binomial Distribution	Poisson Distribution
Nature	Discrete	Discrete
Number of Trials	Fixed (n)	Unlimited
Outcome	Success or Failure	Rare Events
Parameter	Probability of Success (p)	Average Event Rate (λ)
Possible Values	0 to n	0, 1, 2, . . .
Mean	μ = n ⨉ p	μ = λ
Variance	σ2 = n ⨉ p ⨉ (1 – p)	σ2 = λ
Applicability	Limited to a fixed number of trials	Rare events over a large population
Example	Flipping a coin multiple times	Counting occurrences of an event
Assumptions	Independent trials, constant p	Rare events, low probability of success

Poisson Distribution Examples

Example 1: If 4% of the total items made by a factory are defective. Find the probability that less than 2 items are defective in the sample of 50 items.

Solution:  

Here we have, n = 50, p = (4/100) = 0.04, q = (1-p) = 0.96,  λ = 2

Using Poisson’s Distribution,

P(X = 0) = [Tex]\frac{2^0e^{-2}}{0!}[/Tex] = 1/e² = 0.13534

P(X = 1) = [Tex]\frac{2^1e^{-2}}{1!}[/Tex] = 2/e² = 0.27068

Hence the probability that less than 2 items are defective in sample of 50 items is given by:

P( X > 2 ) = P( X = 0 ) + P( X = 1 ) = 0.13534 + 0.27068 = 0.40602

Example 2: If the probability of a bad reaction from medicine is 0.002, determine the chance that out of 1000 persons more than 3 will suffer a bad reaction from medicine.

Solution:

Here we have, n = 1000, p = 0.002, λ = np = 2

X = Number of person suffer a bad reaction 

Using Poisson’s Distribution

P(X > 3) = 1 – {P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)} 

P(X = 0) = [Tex]\frac{2^0e^{-2}}{0!}[/Tex] = 1/e²

P(X = 1) = [Tex] \frac{2^1e^{-2}}{1!}[/Tex] = 2/e² 

P(X = 2) = [Tex] \frac{2^2e^{-2}}{2!}[/Tex] = 2/e²

P(X = 3) =  [Tex]\frac{2^3e^{-2}}{3!}[/Tex] = 4/3e²

P(X > 3) = 1 – [19/3e²] = 1 – 0.85712 = 0.1428

Example 3: If 1% of the total screws made by a factory are defective. Find the probability that less than 3 screws are defective in a sample of 100 screws.

Solution:

Here we have, n = 100, p = 0.01, λ = np = 1

X = Number of defective screws

Using Poisson’s Distribution

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) 

P(X = 0) = [Tex] \frac{1^0e^{-1}}{0!}[/Tex] = 1/e

P(X = 1) = [Tex]\frac{1^1e^{-1}}{1!}[/Tex] =1/e

P(X = 2) = [Tex]\frac{1^2e^{-1}}{2!}[/Tex] =1/2e

P(X < 3) = 1/e + 1/e +1/2e

= 2.5/e = 0.919698

Example 4: If in an industry there is a chance that 5% of the employees will suffer by corona. What is the probability that in a group of 20 employees, more than 3 employees will suffer from the corona?

Solution:

Here we have, n = 20, p = 0.05, λ = np = 1

X = Number of employees who will suffer corona

Using Poisson’s Distribution

P(X > 3) = 1-[P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]

P(X = 0) = [Tex]\frac{1^0e^{-1}}{0!}[/Tex] = 1/e

P(X = 1) = [Tex]\frac{1^1e^{-1}}{1!}[/Tex] = 1/e

P(X = 2) =[Tex]\frac{1^2e^{-1}}{2!}[/Tex] =1/2e

P(X = 3) =[Tex]\frac{1^3e^{-1}}{3!}[/Tex] =1/6e

P(X > 3) = 1 – [1/e + 1/e + 1/2e + 1/6e]

= 1 – [ 8/3e] = 0.018988

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Summary – Poisson Distribution – Definition, Formula, and Table

The Poisson distribution is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space, given a constant average rate of occurrence, λ (lambda), and assuming each event happens independently. It is particularly useful for modeling scenarios where events are rare within the given interval but can happen with a known average frequency. The Poisson distribution is defined by the probability mass function (PMF) P(X = k) = e^−λλ^k / k!, where k represents the number of occurrences, e is the base of the natural logarithm, and λ is both the mean and variance of the distribution. This distribution has several key properties, including a PMF that calculates the likelihood of observing a specific number of events, and it assumes that events occur independently and are distributed uniformly over the interval. The mean and variance of the Poisson distribution are both equal to λ, which simplifies statistical analysis, making it a powerful tool for understanding phenomena in various fields such as physics, biology, and economics.

FAQs on Poisson Distribution – Definition, Formula, and Table

What is poisson distribution?

Probability distribution which is used to model the number of events that occur in a fixed interval of time or space is called Poisson distribution.

When to use poisson distribution?

Poisson Distribution is generally used to represent those events which are seperated over a specific interval of time.

What is poisson distribution expected value?

Expected value is the mean of the Poisson Distribution, and is given by the following formula,

E[X] = λ

What is lambda in poisson distribution?

Lambda is the parameter in Poisson Distribution, which is also equal to mean as well as variance.

What is poisson distribution mean and variance?

Mean is the average value whereas variance is the measure of spread for any data including Poisson Distribution.

When do we use poisson distribution?

Poisson distribution is used to model the number of events or occurrences happening in a fixed interval of time or space when these events are rare and random, and the average rate of occurrence is known.

What is the difference between poisson distribution and normal distribution?

Key difference is in the type of data they model. The Poisson distribution is used for count data representing rare and discrete events, while the normal distribution is used for continuous data representing a wide range of values. Additionally, the Poisson distribution is skewed and discrete, while the normal distribution is symmetric and continuous.

Are the mean and variance of poisson distribution the same?

Yes, in the Poisson distribution, the mean and variance are equal and have the same value, represented by the parameter λ (lambda).