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# Poisson Distribution Formula

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:

f(x) = P(X = x ) = \(\dfrac{e^{ - \lambda } \lambda ^x }{x!}\)

where,

- 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 in the desired time interval
- != factorial of functions

In shorthand notation, it is represented as X ~ P(λ)

Let's see the following properties of a Poisson model:

- The event or success is something that can be counted in whole numbers.
- The probability of having success in a time interval is independent of any of its previous occurrences.
- The average frequency of successes in a unit time interval is known.
- The probability of more than one success in a unit of time is very low.

Let us understand the Poisson distribution formula using solved examples.

## Examples Using Poisson Distribution Formula

Example 1: If the random variable X follows a Poisson distribution with a mean of 3.4, find P(X = 6).

**Solution:**

Given that; X ~ \(P_o\)(3.4)

To find: P(X = 6).

Now, Using the Poisson distribution formula,

P( X = 6) = (e^{-λ} λ^{6} )/6!

= (e^{- 3.4 }× 3.4^{6}) / 6!

= 0.071604409

= 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.

**Solution:**

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 }× 1.2^{2} )/ 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?

Solution:

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} 10^{10})/ 10!

=(4.54 10^{-5}10^{10})/3528800

=4540/36288

= 0.12511

**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} λ^{x})/ x!

Where

- 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} λ^{x})/ 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} λ^{x})/ x!. Thus we calculate P(getting 4 floods) = (e^{-3} 3^{4 })4!= 0.168 = 17%

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