Why are geometric distributions called geometric

Geometric distribution

This article explains in a simple and understandable way what the geometric distribution has on itself. You will learn when to use this discrete distribution and how to use it Formulas to calculate the Expected value, the density and theDistribution function used.

You want to know what the geometric distribution is with Bernoulli has to do without reading this article? Then check out our Video on the topic!

Geometric distribution statistics

The geometric distribution is a discrete probability distribution, which is based on independent Bernoulli Experiments.

It is often referred to as "distributing the waiting for the first success". Their basic idea is also reflected in this expression. It is a Bernoulli's experiment with the probability p and one wonders how often one has to carry out this experiment before the first success occurs.

Geometric distribution example

A classic example of this is “Don't get angry”. In order to be allowed to go onto the playing field with your piece, you have to roll six dice. Now one may wonder how many times you have to roll the die to get a six. Every attempt to roll a six is ​​a Bernoulli experiment with the probability p = .
Mathematically, the geometric distribution is expressed as follows:

X ~ G (p)

Or in our example:

X ~ G (

Geometric distribution density

The densitythe geometric distribution is as follows:

You can easily deduce yourself why the formula looks like this. For example, if you want to calculate the probability that you will roll a 6 in the second attempt, i.e. f (2), you have to calculate the probability that you did not roll a 6 in the first attempt, i.e. (1-p), with the probability that you roll a 6, i.e. p, multiply. The probability of rolling a 6 in the second attempt is therefore around 13.89%.

Geometric distribution distribution function

The easiest way to determine the distribution function of the geometric distribution is to use the counter-probability. We are looking for the probability that fewer than x attempts are needed to roll a 6. To do this, you would have to add up each individual probability. But you can save yourself that by simply determining the probability for the counter-event, i.e. P (X> x). Because the rule applies that the probability is 1 minus the opposite probability.

Expected value of geometric distribution

The expected value of the geometric distribution can also be determined very easily:

With a probability of p = , so you need an average of 6 attempts to roll a 6.

Geometric distribution variance

If you know the expected value, calculating the variance is easy. The formula for determining the variance is as follows:

In the end, you just have to insert the expected value and get the variance you are looking for in no time at all.

Geometric distribution formula

Excellent! That's it for the geometric distribution! Here are all the important formulas summarized again: