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 × λx) / 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 × λx) / 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