# Poisson distribution

If your provider has interruptions for an average of 20 minutes a week, what is the probability that there will be no Internet for an hour today? 22.3%! But for two hours we will be left without Internet with a probability of only 4.3%. So, the Poisson distribution is just for this, knowing the average value, we can get the probability that the event will happen for the period we are interested in.

## Probability (in general)

A very important probability, used literally every day in production and in the service sector and other sciences.
The essence is very simple: if we know the probability of an event and if such events occur __independently__ of each other, then
we can find out:

- a. The probability that N events will occur
- b. The probability that fewer or more than N events will occur

## Probability (specific case)

### Example

The plant produces 1000 meters of cable per day, at a cost of 500 rubles per 1 m. On average, one marriage is detected per 300 meters of produced and then the meter of the wire is cut off. What is the probability that the plant will lose more than 7000 rubles in three days?

The probability that the wire will be defective is 1/300 ~ 0.0033.

The loss of the plant by 7000 rubles is 14 defective meters.

The lambda parameter for three days is: λ 3000*0.0033 = 10.

The cumulative value of the Poisson distribution for & lambda; = 10 is equal to F(14) = 0.9165, from where the probability of getting more than 14 defective meters in three days is 1-0.9165 = 0.08835 = 8.3%.The main task of such distributions is the ability to predict losses, make plans for the future.

## Poisson distribution - definition

The Poisson distribution is a discrete type probabilistic distribution that models a random variable representing the number of events, occurred in a fixed time, provided that these events occur with some fixed average intensity and independently from each other. In other words, if an event occurs with some frequency, then we can determine the probability that such an event it will happen n times during the period we are interested in.

### Lambda parameter - λ

The Poisson distribution depends on only one parameter - λ, this parameter depends
on the probability of a successful event and the total number of events.

Successful event: The Poisson distribution is applied only when there is a division into the result "yes" and "no", for example, a light bulb burned out:
yes - successful event; tire punctured: yes - successful event and so on.

λ = n*p, where p is the probability of a successful event, and n is the total number of events for which the calculation is performed.

For example, if a thunderstorm passes once a month and we want to calculate the probability of a thunderstorm for 24 months, then the probability is one, and the number of events is 24, from where lambda is 24.

It can be considered differently, the probability of a thunderstorm on a particular day is 1/30, the number of events is 730 days, lambda is 24.3.

#### Example

In a thousand boxes with antonovki in one comes across golden, what is the probability that in 5000 boxes there will be less than 4 boxes of golden apple?

The probability of a box with a golden apple is 0.1% (1 box per 1000 = 1/1000, if the percentage is 1/1000 * 100 = 0.1%)

Total number of events - 5000 boxes

From the above it follows:

**λ = 5000 * 0.001 = 5**

From where the probability is 26.5% (calculator below).

### Probability function (Poisson formula)

The probability that a successful event will occur k times:

f(k) = P(k) = λ^{k}* e^{-λ}/ k!

#### Example

In a thousand boxes with antonovki in one comes across golden, what is the probability that in 5000 boxes there will be 2 boxes with golden apple?

From the previous example, we know that λ =5, now we are looking for the probability that k will be equal to 2, for this we use the formula of the probability function:

f(4) = P(k = 4) = λ^{k}e^{-λ}/ k! = 5^{2}* e^{-5}/ 2! = 0.084 = 8.4%

### Distribution law

F(n) = P(k ≤ n) = G (k+1,λ) / k!

## Poisson distribution graph

Note that when you increase λ, the distribution graph becomes similar on normal distribution graph.

## Poisson distribution in excel

The function for finding the value of the Poisson distribution in excel is called "POISSON". The function accepts two required parameters - number and average, as well as an optional parameter "Integral value", so to get the value of the probability function P(≤n), enter the formula:

=POISSON(n;λ)

=POISSON(n;λ)

If it is necessary to get a point value, i.e. only for n:

=POISSON(n;λ;FALSE) =POISSON(n;λ,FALSE)

Example: for a distribution with the parameter λ=5, find out the probability of the origin of three events:

=POISSON(3;5;FALSE) =POISSON(3;5;FALSE)