Poisson Distribution Calculator

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