Poisson Distribution Calculator

Poisson Distribution Calculator

Poisson distribution calculator is an online tool that calculates probabilities of any event for specific intervals with the poisson distribution formula. Our Poisson calculator offers different probability type options to find the likelihood value of a poisson random variable.

This poisson probability calculator shows the poisson distribution table of the probability of random variables for given interval values. It also converts the poisson distribution result into poisson and cumulative distribution percentages for easy interpretation.

What is Poisson Distribution?

The poisson distribution is aka discrete probability distribution that expresses the probability or likelihood of a number of events occurring in a fixed time by performing a poisson experiment. It was named on behalf of French mathematician Simeon Denis Poisson, who gave a remarkable idea for probability using a discrete probability distribution.

Poisson distribution shows the probability of any event if the rate of success is small and the number of trials is large. The occurrence of any event depends on the average rate of success but is independent with respect to time. For the likelihood value of any event use the poisson distribution formula.  

Poisson Distribution Formula

The poisson distribution formulas for the different conditions with random variable “X” and average rate “λ” in the mathematical form are stated as:

For exactly “X” values:

P(X= x) = e^-λ * λ^x/x!

For More than “X” values:

P(X > x) = 1 - P(X ≤ x)

For Less than “X” values:

P(X < x) = Σ^x-1_k=0 P(k) 

For At most “X” values:

P(X ≤ x) = Σ^x_k=0 P(k) 

For At least “X” values:

P(X ≥ x) = 1 - P(X < x)

Breakdown of the poisson formulas:

• “e” is Euler’s number (approximately 2.71828).
• “X” is the total number of occurred events.
• “λ” is a number of favorable events.
• “X!” is the factorial of X.

These formulas help to calculate the probability of Poisson distribution manually. For a quick solution use our poisson distribution calculator, which provides the accurate probability value in a matter of seconds. 

How to Calculate Poisson Distribution?

To find the Poisson distribution value manually by using the poisson formula follow the below simple steps:

• Note the parameter lambda “λ” values by analyzing data.
• Choose the poisson random variable “X” (must be a non-negative integer), that is used calculate the probability.
• Lastly, put all values in the poisson distribution formula and get results according to the types of probability.

Alternatively, use poisson distribution calculator to avoid all the above steps and get accurate results with a single click. It provides step-by-step solutions to each problem with 100% accuracy.

Poisson Distribution Examples

In this section, describe the method to calculate the probability of events using the Poisson distribution method and its formula with steps. This step helps to understand the calculation of poisson values easily and in a better way. 

Example 1

Find the probability of the poisson variable at exactly “3” random variable value when the average rate (λ) is 2. 

Solution:

Step 1: Note data from the given statement.

λ=2,    X =3

Step 2: Calculate the values of “x!”, “e-λ” and “λx”.

e^-λ = e^-2 = 0.1353

λx = 2³   = 8

x! = 3!  = 6

Step 3: Now, Substitute all values in the poisson distribution formula.

P (X = 3) = 0.1353 × 8 / 6 = 1.0824 / 6 = 0.1804

Thus, the percentage probability is “18.04%” on exactly “3” random variable values. 

Example 2

Calculate the probability for all possibilities if the “4” calls done in an hour and the average rate of success of a call is “3” in one hour. 

Solution:

Step 1: Note the value of “X” and “λ” from given data.

λ=3,    X =4

The possibilities of probability are “Exactly X, More than X, Less than X, At most X, and At least X”.

Step 2:  Use the formula of each possibility and get the values for it.

For Exactly “X” values:

P(X = 4) = (e^-3 * 3⁴)/4! 

P(4) = (0.0498 * 81)/24

P(4) = 4.0338/24

P(4) = 0.1681

For Less than “X” values:

P(X < 4) = Σ^4-1_k=0 P(k) = P(0) + P (1)+ P(2) + P (3)

P(0) = (e^-3 * 3⁰)/0! = 0.0498

P(1) = (e^-3 * 3¹)/1! = 0.1494

P(2) = (e^-3 * 3²)/2! = 0.2241

P(3) = (e-3 * 3³)/3! = 0.2241

P(X < 4) = Σ^4-1k=0 P(k) = P(0) + P (1)+ P(2) + P (3)

= 0.0498 + 0.1494 + 0.2241 +  0.2241

P(X < 4) = 0.6474

For At most “X” values:

P(X ≤ 4) = Σ⁴_k=0 P(k) = P(0) + P (1)+ P(2) + P (3) + P(4)

P(0) = (e^-3 * 3⁰)/0! = 0.0498

P(1) = (e^-3 * 3¹)/1! = 0.1494

P(2) = (e^-3 * 3²)/2! = 0.2241

P(3) = (e^-3 * 3³)/3! = 0.2241

P(4) = (e^-3 * 3⁴)/4! = 0.1681

P(X ≤ 4) = Σ⁴_k=0 P(k) = P(0) + P (1)+ P(2) + P (3) + P(4)

= (0.0498) + (0.1494) + (0.2241) + (0.2241) + (0.1681)        

P(X ≤ 4) = 0.8155

For More than “X” values:

P(X > x) = 1 - P(X ≤ x)

P(X > 4) = 1 - P(X ≤ 4)

= 1 - 0.8155 

P(X > 4) = 0.1845

For At least “X” values:

P(X ≥ x) = 1 - P(X < x)

P(X ≥ 4) = 1 - P(X < 4)

= 1 - 0.6474 

P(X ≥ 4)= 0.3526

To verify the results of all the above probabilities use our poisson calculator, which provides an accurate answer with detailed steps.

Frequently Asked Questions

What is lambda in Poisson distribution?

The lambda (λ) is the mean or average rate that shows the occurrence of events within a specific interval of time or space. It’s also interpreted as the rate of success of occurrence of any event.

When to use Poisson Distribution?

A Poisson distribution finds the likelihood of a specific number of events happening in a given interval of time or space. The poisson distribution is used when the average rate of occurrence is constant over time and events occur independently. 

Can Poisson have a negative mean?

No, Poisson distribution finds the value of zero and positive integers then it can’t give the negative values.

Are Poisson distribution and binomial distribution the same?

No, binomial distribution describes the distribution of binary data from a finite sample. In contrast, poisson distribution describes the distribution of binary data from an infinite sample.

Is Poisson continuous or discrete distribution?

The poisson distribution is also called the discrete probability distribution because it finds the probability or likelihood of a specific number of events for a specific time.