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.

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, Poisson Distribution formula, Poisson Distribution examples, and properties of Poisson Distribution in detail.

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 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 Probability of Observing k Events
• e is Base of Natural Logarithm (approximately 2.71828)
• λ is Average Rate of Occurrence of Events
• k is Number of Events that Occur

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.

• 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.

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

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 Mean of Poisson’s Distribution
• λ is Parameter of Distribution
• X is 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 Number of Trails
• p is 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 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 Average rate of Occurrence of Events
• σ (sigma) is Standard Deviation of Distribution

Probability Mass Function for Poisson Distribution is given by:

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

where,

• λ is Parameter which is also equal to Mean and Variance
• k is Number of times an event occurs
• e is Euler’s Number (≈2.718)

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

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

Q1. A call center receives an average of 5 calls per minute. What is the probability that exactly 7 calls are received in a minute?

Q2. On average, a person receives 3 emails per hour. What is the probability that the person receives no emails in a given hour?

Q3. A factory produces an average of 2 defective products per day. What is the probability that exactly 3 defective products are produced in a day?

Q4. A store experiences an average of 10 customer arrivals per hour. What is the probability that exactly 15 customers arrive in an hour?

Q5. A machine has an average failure rate of 1 failure per month. What is the probability that the machine will not fail at all in a given month?

Important Maths Related Links:

• Area of Quadrant
• Difference Between Ln And Log
• Simple Interest Rate Formula
• Maths Project Work
• Table of 17
• Area of A Circle Formula
• Congruent Figures
• Semi Circle Formula
• Table of 9
• Three Types of Triangles

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.

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 Difference between Poisson Distribution and Normal Distribution?

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.

Are Mean and Variance of Poisson Distribution Same?

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