In probability theory, the Poisson distribution is a very common discrete probability distribution. A Poisson distribution helps in describing the chances of occurrence of a number of events in some given time interval or given space conditionally that the value of average number of occurrence of the event is known. This is a major and only condition of Poisson distribution.

The distribution that arises from the Poisson experiment is termed as Poisson distribution. The number of successes that are resulting from a Poisson experiment is termed as a Poisson random variable.

**1)** The outcomes of the experiment can be easily classified as either success or failure.

**2)** The average of the number of successes within a region that is specified is known.

**3)** The probability of occurrence of a success is always proportional to the size of the specified region.

**4)** The probability of occurrence of success in a very small region is zero virtually.

An experiment in statistics is termed as Poisson experiment when it possesses the following probabilities:

It is to be noted that the region that is specified can take different forms like area, length, time period etc.

P (x, $\mu$) = $\frac{(e^{-\mu}) (\mu^x)}{x!}$

Here, ‘x’ represents the actual number of occurring successes that are resulting from the Poisson experiment, the value is ‘e’ is 2.71828 approximately, '$\mu$' is the average of the number of successes that are within a specified region.

$\mu$= 3, average number of files completed a day

x = 5, the number of files required to be completed next day

And e = 2.71828 being a constant

On substituting the values in the Poisson distribution formula mentioned above we get the Poisson probability in this case.

We get,

P(x,$\mu$) = $\frac{(e^{-\mu}) (\mu^x)}{x!}$

$\rightarrow$ P (5, 3) = $\frac{(2.71828)^{-3} (3^5)}{5!}$

= 0.1008 approximately.

Hence the probability for the person to complete 5 files the next day is 0.1008 approximately.

**Example 2:** If customers come into a bank with variance 36/hour. Find the standard deviation of customer visit per hour using Poisson distribution.

**Solution:** According to a Poisson distribution, the expected value is $\mu_x$ = variance = $\lambda$ = 36 customers per hour.

Now the standard deviation = $\sigma$ = $\sqrt{\lambda}$ ($\because$ standard deviation = square root of variance)

$\sigma$ = $\sqrt{36}$ = 6 customers per hour.

Now the standard deviation = $\sigma$ = $\sqrt{\lambda}$ ($\because$ standard deviation = square root of variance)

$\sigma$ = $\sqrt{36}$ = 6 customers per hour.