A Poisson experiment is a statistical experiment and a theoretical discrete probability that classifies the experiment into two categories, success or failure. Poisson distribution is a limiting process of the binomial distribution. The Poisson distribution formula is used to find the probability of events happening when we know how often the event has occurred. Let us understand the Poisson distribution formula using solved examples.
What Is Poisson Distribution Formula?
Poisson Distribution Formula is used to show the number of times an event is likely to occur within a specified time duration. A Poisson random variable “x” is used to define the number of successes in the experiment. This distribution generally models the number of independent events within the given time interval. The Poisson distribution formula is very useful in situations where discrete events occur in a continuous manner.
The Poisson distribution is used under certain conditions:
The number of trials, n, tends to infinity
The probability of success, P, tends to zero
np = 1 is finite
For a Poisson random variable, x = 0,1,2, 3,.........∞, the Poisson distribution formula is given by:
Example 1: If the random variable X follows a Poisson distribution with a mean of 3.4, find P(X = 6).
Given that; X ~ \(P_o\)(3.4)
To find: P(X = 6).
Now, Using the Poisson distribution formula,
P( X = 6) = (e-λ λ6 )/6!
= (e- 3.4 × 3.46) / 6!
= 0.072 Answer: P(X = 6) = 0.072
Example 2: A factory produces nails and packs them in boxes of 200. If the probability that a nail is substandard is 0.006, find the probability that a box selected at random contains at most two nails that are substandard. Use the Poisson distribution formula.
If X is the number of substandard nails in a box of 200, then
X ~ B (200,0.006)
Since n is large and p is small, the Poisson approximation can be used.
The appropriate value of λ is given by
λ= np = 200 × 0.006 = 1.2
So, X ~ \(P_o\) (1.2) and
P(X≤ 2) = e - 1.2 + e- 1.2 × 1.2 + (e- 1.2 × 1.22 )/ 2!
= 2.92 e - 1.2
= 0.8795 Answer: The probability that a box selected at random contains at most two nails that are substandard is 0.8795
Example 3: The average number of cars passing through a tunnel per minute is 5. What is the probability that the expected number of cars actually pass through in a given 2 minute-time?
Given: λ = 5.
At 2 minutes λ = 10.
To find: P(X = 10).
Using the Poisson distribution formula, we have
P(X=10) = (e -10 1010)/ 10!
Answer: Thus the probability of the number of cars passing through a 2 minute-time is 0.12511
FAQs on Poisson Distribution Formula
What is the Poisson Distribution formula in Statistics?
The Poisson distribution formula is used to find the probability of events happening when we know how often the event has occurred. For a Poisson random variable, x = 0,1,2, 3,.........∞, the Poisson distribution formula is given by:
P(X=x)= (e -λ λx)/ x!
e is the Euler's number(e = 2.71828)
x is a Poisson random variable that gives the number of occurrences(x= 0,1,2,.......)
λ is an average rate of value or the expected number of occurrences
!= factorial of functions
When to Use Poisson Distribution Formula?
When the average probability of an event happening per time period is known and we are about to find the probability of a certain number of events happening in the time period, we use the Poisson distribution. We model the Poisson distribution of rare events in a large population.
How is Poisson Calculated in The Poisson Distribution Formula?
When the mean number of successes(λ) is known and e = 2.71828, we find the number of successes(x) out of the experiment conducted. We can find the probability of the number of successes by choosing a Poisson random variable. We apply these values in the formula, P(X=x)= (e -λ λx)/ x!
What Are The Applications of the Poisson Distribution Formula?
The Poisson distribution formula is used in situations where discrete events occur in a continuous manner. It is used by scientists and businessmen to forecast weather, the sale in a year, the average number of customers in a week or month, and so on. For example, the number of floods per year in the country is 3. The probability that exactly 4 floods will affect the country next year is given by applying the Poisson distribution formula: P(X=x)= (e -λ λx)/ x!. Thus we calculate P(getting 4 floods) = (e-3 34 )4!= 0.168 = 17%