What is Poisson Distribution in Probability – Definition, Formulas & Examples

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Over many years, statisticians noticed that data from samples and populations often formed very similar patterns. For example, a lot of data were grouped around the ‘middle’ values, with fewer observations at the outside edges of the distribution (very high or very low values). These patterns are known as ‘distributions’, because they describe how the data are ‘distributed’ across the range of possible values.

There are many types of distributions in statistics suited for some particular application or environment. Some of the most common types of distributions in statistics are Normal or Gaussian distribution, Bernoulli Distribution, Binomial distribution, Poisson distribution, Exponential distribution, Gamma distribution, and Weibull distribution.

Let’s understand what is Poisson distribution, and the formulas used with examples.

What is Poisson Distribution in Probability?

Poisson distribution named after the French mathematician Denis Poisson is a theoretical discrete probability. It is also known as the Poisson distribution probability mass function. Poisson distribution is used to find the probability of an independent event occurring in a fixed interval of time with a constant mean rate. In other words, Poisson distribution is used to estimate how many times an event is likely to occur within a given period of time. $\lambda$ is the Poisson rate parameter that indicates the expected value of the average number of events in the fixed time interval. Poisson distribution has wide use in the fields of business as well as in biology.

The Poisson distribution is used as a limiting case of the binomial distribution when the trials are large indefinitely. If a Poisson distribution models the same binomial phenomenon, $\lambda$ is replaced by $np$. 

Examples of Poisson Distribution

In general, Poisson distributions are often appropriate for count data. Count data is composed of observations that are non-negative integers (i.e., numbers that are used for counting, such as 0, 1, 2, 3, 4, and so on). 

The following are some examples of Poisson distribution.

• Text messages per hour
• Cattle per hectare
• Machine malfunctions per year
• Website visitors per month
• Influenza cases per year

Properties of Poisson Distribution

The Poisson distribution is applicable in events that have a large number of rare and independent possible events. The following are the properties of the Poisson Distribution. 

• The events are independent.
• The average number of successes in the given period of time alone can occur. No two events can occur at the same time.
• The Poisson distribution is limited when the number of trials n is indefinitely large.
• mean = variance = $\lambda$
• $np = \lambda$ is finite, where $\lambda$ is constant.
• The standard deviation is always equal to the square root of the mean $\mu$.
• If the mean is large, then the Poisson distribution is approximately a normal distribution.

When Should We Use Poisson Distribution?

You can use a Poisson distribution to predict or explain the number of events occurring within a given interval of time or space. “Events” could be anything from disease cases to customer purchases to meteor strikes. The interval can be any specific amount of time or space, such as $10$ days or $5$ square inches.

You can use a Poisson distribution if:

• Individual events happen at random and independently. That is, the probability of one event doesn’t affect the probability of another event.
• You know the mean number of events occurring within a given interval of time or space. This number is called $\lambda$, and it is assumed to be constant.

When events follow a Poisson distribution, $\lambda$ is the only thing you need to know to calculate the probability of an event occurring a certain number of times.

What is Poisson Distribution Formula? 

The Poisson distribution formula is used to find the probability of an event that happens independently, discretely over a fixed time period, when the mean rate of occurrence is constant over time. The Poisson distribution formula is applied when there is a large number of possible outcomes. 

For a random discrete variable, $\text{X}$ that follows the Poisson distribution, and $\lambda$ is the average rate of value, then the probability of $x$ is given by $f(x) = \text{P}(\text{X}=x) = \frac{e^{-\lambda} \lambda^ x }{x!}$

where

$x = 0, 1, 2, 3…$

$e$ is the Euler’s number($e = 2.718$)

$\lambda$ is an average rate of the expected value and $\lambda$ = variance, also $\lambda > 0$

Examples on Poisson Distribution Formula

Example 1: In a cafe, the customer arrives at a mean rate of $2$ per min. Find the probability of the arrival of $5$ customers in $1$ minute assuming that customer arrival follows a Poisson distribution.

Mean $\lambda  = 2$, 

and $x = 5$

Using the Poisson distribution formula $\text{P}(\text{X}=x) = \frac{e^{-\lambda} \lambda^ x }{x!}$

$\text{P}(\text{X} = 5) = \frac{e^{-2} 2^5}{5!}$

$=> \text{P}(\text{X} = 6) = 0.036$

Thus, the probability of the arrival of $5$ customers per minute is $3.6\%$.

Example 2: If $3\%$ of electronic units manufactured by a company are defective. Find the probability that in a sample of $200$ units, less than $2$ bulbs are defective.

The probability of defective units $p = \frac{3}{100} = 0.03$

$n = 200$

We observe that $p$ is small and $n$ is large here. Thus it is a Poisson distribution.

Mean $\lambda = np = 200 \times 0.03 = 6$

$\text{P}(\text{X}= x)$ is given by the Poisson Distribution Formula as $\frac{e^{- \lambda} \lambda ^x }{x!}$

$\text{P}(\text{X} < 2) = \text{P}(\text{X} = 0) + \text{P}(\text{X}= 1)$

$=\frac{e^{-6} 6^0 }{0!} + \frac{e^{-6} 6^1}{1!}$

$= e^{-6} + e^{-6} \times 6$

$= 0.00247 + 0.0148$

$\text{P}(\text{X} < 2) = 0.01727$

Thus the probability that less than $2$ bulbs are defective is $0.01727$.

Example 3: The number of meteorites that hit the earth in a given day is modeled by a Poisson Distribution with 

$\lambda = 4$. What is the probability that $5$ meteoroids hit the earth in a day?

Mean $\lambda  = 4$, 

and $x = 5$

Using the Poisson distribution formula $\text{P}(\text{X}=x) = \frac{e^{-\lambda} \lambda^ x }{x!}$

$\text{P}(\text{X} = 5) = \frac{e^{-4} 4^5}{5!}$

$= \frac{0.01831563888 \times 1024}{120} = 0.156$

Thus, the probability $5$ meteoroids hit the earth in a day = $ 0.156$

Poisson Distribution Mean and Variance

For the Poisson distribution, which has $\lambda$ as the average rate, for a fixed interval of time, then the mean of the Poisson distribution and the value of variance will be the same. So for $\text{X}$ following Poisson distribution, we can say that $\lambda$ is the mean as well as the variance of the distribution.

Hence, $\text{E}(\text{X}) = \text{V}(\text{X}) = \lambda$

where

$\text{E}(\text{X})$ is the expected mean

$\text{V}(\text{X})$ is the variance

$\lambda > 0$

Applications of Poisson Distribution

Poisson distribution is used in wide areas. The random variables that follow a Poisson distribution are as follows:

• To count the number of defects in a finished product
• To count the number of deaths in a country by any disease or natural calamity
• To count the number of infected plants in the field
• To count the number of bacteria in the organisms or the radioactive decay in atoms
• To calculate the waiting time between the events

Practice Problems

1. At a small walk-in clinic, an average of five patients arrive at the clinic per hour during opening hours. What is the probability that exactly three patients will arrive in the next hour? Assume that the number of patients arriving per hour follows a Poisson distribution.
2. If you receive an average of two emails per week from your statistics professor, what is the probability that you will receive exactly one email from your statistics professor on Monday? Assume that the number of emails per day follows a Poisson distribution.
3. Over the last $300$ years, there were $87$ floods in an area. Assuming that the number of floods per year follows a Poisson distribution, what is the probability that there will be no floods in that area next year?
4. Suppose the average number of major storms in an area is $4$ per year. What is the probability that exactly $7$ storms will hit the area next $2$ years?
5. If births in a hospital occur randomly at an average rate of $1.8$ births per hour, what is the probability of observing $4$ births in a given hour at the hospital?
6. The random variable $x$  is the number of occurrences of an event over an interval of ten minutes. It can be assumed that the probability of an occurrence is the same in any two-time periods of equal length. It is known that the mean number of occurrences in ten minutes is $5.3$. What is the expected value of the random variable $x$?

FAQs

Conclusion

Poisson distribution is used to find the probability of an independent event occurring in a fixed interval of time with a constant mean rate. For a random discrete variable, $\text{X}$ that follows the Poisson distribution, and $\lambda$ is the average rate of value, then the probability of $x$ is given by $f(x) = \text{P}(\text{X}=x) = \frac{e^{-\lambda} \lambda^ x }{x!}$, where $x = 0, 1, 2, 3…$, $e$ is the Euler’s number($e = 2.718$) and $\lambda$ is an average rate of the expected value.

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