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.

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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.

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).

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

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

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

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

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} × λ^{x}) / 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**(**x**≥**1**)

P(x ≥ 2) = 1 - P(x < 2)

P(x≥1) = 0.99326

Poisson Distribution - an overview | ScienceDirect Topics