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# Poisson Distribution Calculator

## How to use Poisson distribution calculator?

By following the below steps you can evaluate the probability of the given data:
• Enter the “Average rate of occurrences (λ)”
• Enter the “Poisson random variable (x)”
• Click on the “calculate” button to evaluate the result.
• You can erase all inputs by clicking the “reset” button.
• Click on the “show steps” button to see the step-by-step solution.

## Poisson distribution calculator:

The Poisson distribution calculator calculates the probability of the given data, it calculates all the possible probabilities according to the rules of the Poisson distribution. I.e., For Exactly, For Less than, For At most, For More than, For At least.

## What is Poisson distribution?

In the probability theory, the Poisson distribution is used to calculate the discrete probability, as a result, it gives the probability of an event occurring multiple times (x).

## Formula of Poisson distribution:

• “e” is Euler’s constant
• “λ” is the average rate of occurrence
• “x” is the Poisson random variable

## Properties of the Poisson distribution:

Just like the other discrete probability distributions, the Poisson distribution also has some properties that are as follows:

### Mean of Poisson distribution:

The mean of the Poisson distribution is given by lambda (λ).

### Standard deviation:

The standard deviation of the Poisson distribution is the square root of the mean given by √(λ).

## Examples of Poisson distribution:

This section explains the method of evaluating probability using the Poisson distribution.

Example 1:

Calculate all the possible probabilities if the average rate of occurrence is 5 and the Poisson random variable is 1.

Solution:

Step 1: Extract the data.

Average rate of occurrence = λ = 5

Poisson random variable = x = 1

Euler’s constant = e = 2.718

Step 2: Calculation

For exactly P (x = 1):

Formula

P(x) = (e−λ × λx) / x!​

Values:

e = 2.718

x = 1

λ = 5

For Probability: P(x = 1)

P(1) = {(2.718) −(5) × (5)1​} / 1!

P(1) = 0.00674 × 5

​P(x = 1) = 0.03371

For Probability: P(x < 1)

P(0) = {(2.718) −(5) × (5)0} / 0!

​P(0) = 0.00674

P(x < 2) = P (0)

P(x<1) = 0.00674

For Probability: P(x 1)

P(0) = {(2.718)−(5) × (5)0} / 0!

​P(0) = 0.00674

P(1) = {(2.718)−(5)×(5)1} / 1!​

P(1) = 0.03371

P(x ≤ 2) = P(0) +P(1)

P(x≤1) = 0.04045

For Probability: P(x>1)

P(x > 2) = 1 - P(x ≤ 2)

P(x>1) = 0.95955

For Probability: P(x1)

P(x ≥ 2) = 1 - P(x < 2)

P(x≥1) = 0.99326

## References:

Poisson Distribution - an overview | ScienceDirect Topics