# How is the Poisson distribution used

## Poisson distribution

You can get them herePosisson distribution explained simply and using an example. We'll show youformula for the density andCalculation tips the distribution function, the expected value & the variance. In short, this summary contains everything you need to know about the Poisson distribution.

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### Poisson distribution statistics

The Poisson distribution is one of the discrete distributions. It is mainly used when the frequency of an event over a certain period of time is examined in a random experiment.

In mathematical terms, the Poisson distribution looks like this:

Lamda stands for the average number of events that can be expected.

### Poisson distribution example

In everyday life there are countless situations that can be calculated with the help of the Poisson distribution. generally this is used when n is very large and p is small. In this case, the calculation of the binomial coefficient would be very time-consuming. Thus, the Poisson distribution serves to approximate the more complicated oneBinomial distribution.

An example of this would be the question of how many students come into the lecture hall between 12:00 and 12:15. Consequently, the expected number of students is sought.

If we assume that an average of 10 students enter the lecture between 12.00 and 12.15, we would write it down as follows:

### Poisson distribution formula

To calculate the probability of the random variable you are looking for, you should memorize the following formulas and relationships.

### Poisson distribution density

The formula for that densityIn this context it looks a bit uncomfortable, but it is actually not very complicated:

In our example, this could be used to calculate the probability that exactly 12 students will enter the lecture hall between 12 p.m. and 12.15 p.m. To do this, just put x equal to 12 and lamda equal to 10 into the equation. You get a probability of about 9.5%.

### Poisson distribution distribution function

Unfortunately, there is once again no convenient formula for the distribution function. You have to add up the individual values ​​of the density function here:

### Poisson distribution expectation

The expected value of the Poisson distribution is very easy to determine: it is simply described by the value lamda. That is also logical, since the expected value describes the expected value and lamda expresses exactly that. Sometimes this is also noted with the small Greek letter µ. Among other things, the expected value in this context can also be referred to as an intensity parameter.

### Poisson distribution variance

The Standard deviation σ and variance σ² are calculated directly as usual with the help of the expected value. The Poisson distribution variance again corresponds to the value lamda. If we look at the standard deviation, this results logically from the root of the expected value.

Now the Poisson distribution is no longer a problem for you!