## What is Poisson distribution?

Poisson distribution is a theoretical distribution that is a good approximation to the binomial distribution when the probability is small and the number of trials is large.

### Formula:

**P(x) = (e**^{−λ} × λ^{x}) / x!

- “
**e**” is Euler’s constant (**e = 2.718**). - “
**λ**” expected number of events (average rate of occurrence). - “
**X**” observed number of events (Poisson random variable).

## Example section:

In this section, we'll learn the steps for calculating the probability using the Poisson distribution method.

**Example 1:**

If **x = 2**(Poisson random variable) is an observed number of events and **λ = 2** (average rate of occurrence) expected number of events find the possible probabilities.

**Solution:**

**Step 1: **Extract the data

X = 2

λ = 2

e =2.718

**Step 2:**Find the probability

**x = 2** **(For Exactly)**

Formula:

P(x) = (e^{−λ} × λ^{x}) / x!

Values:

e = 2.178, x = 2, λ = 2

for probability: P (x = 2)

P (2) = {(2.718)^{ −(2)} × (2)^{2}} / 2!

P (2) = 0.54144 / 2

**P (x = 2) = 0.27073**

**For probability: P (x < 2) (For less than)**

P (0) = {(2.718)^{ −(2)} × (2)^{0}} / 0!

P (0) = 0.13536

P (1) = {(2.718)^{ −(2)} × (2)^{1}} / 1!

P (1) = 0.27073

P (x < 2) = P (0) + P (1)

**P (x < 2) = 0.40609**

**For Probability: P (x ≤ 2) **

P (0) = {(2.718)^{ −(2)} × (2)^{0}} / 0!

P (0) = 0.13536

P (1) = {(2.718)^{ −(2)} × (2)^{1}} / 1!

P (1) = 0.27073

P (2) = {(2.718)^{ −(2)} × (2)^{2}} / 2!

P (2) = 0.54144 / 2

P (x = 2) = 0.27073

P (x ≤ 2) = P (0) + P (1) + p (2)

**P (x ≤ 2) = 0.67682**