Geometric distribution

The geometric distribution is a discrete probability distribution for independent Bernoulli trials. It defines two variants:

The two variants are in the relationship. Which of these is called " geometric distribution ", is either specified in advance or you can choose the one that is just more convenient.

The geometric distribution is used:

  • The analysis of the waiting times until the arrival of a particular event. in the lifetime determination of equipment and components, namely the wait -to-failure
  • Determine the reliability of devices ( MTBF )
  • Determining the risk in the actuarial
  • Determining the error rate in data transmission, eg the number of successfully transmitted TCP packets between two packets with Retransmission
  • 3.1 Relationship to the negative binomial distribution
  • 3.2 Relationship to the exponential distribution

Definition of the geometric distribution

A discrete random variable or with the parameter ( probability of success ), ( probability of failure ) satisfies the geometric distribution if:

In both cases the values ​​for the probabilities form a geometric sequence.

So that the geometric distribution has the following distribution functions


Expected value

The expectation values ​​of the two geometric distributions are

The expected value may be derived in several ways:

  • .
  • The expected value can be broken down by case distinction. With probability of the first experiment is successful, that is, it is realized with 1. With probability the first experiment is unsuccessful, but the expectation value for the number of still following experiments is due to the memoryless again. So true
  • Leads to experiments, it is the expectation value for the number of successful experiments. Therefore, the expected distance between two successful experiments (including a successful experiment ), ie.


The variances of the two geometric distributions are

The derivation can be effected via


The geometric distribution is a memoryless distribution, ie it applies to

Option A

Variant B

So is of a random variable geometrically distributed announced that it is greater than the value ( variant A ) or at least has the value (variant B), then the probability that it exceeds this value to, the same size as the one that an identical random variable takes the value at all.

The memoryless is a defining characteristic; the geometric distribution is therefore the only possible memoryless discrete distribution. Your steady counterpart here is the exponential distribution.

Relation to reproductive

The sum of independent geometrically distributed random variables with the same parameter is not geometrically distributed, but negative binomial distribution. Thus, the family of geometric probability distributions is not reproductive.


The skewness is given by:


The curvature can also be represented as a closed

Characteristic function

The characteristic feature is in the form

Moment generating function

The moment generating function of the geometric distribution is

Relations with other distributions

Relationship to the negative binomial distribution

A generalization of the geometric distribution represents the negative binomial distribution, which indicates the probability that, for successful attempts are necessary, or ( in an alternative representation ) that the -th success occurs after failure already occurred.

Conversely, the geometric distribution is at a negative binomial distribution.

Relationship to the exponential distribution

For a sequence of geometrically distributed random variables with parameters applies to a positive constant. Then the sequence for great against an exponentially distributed random variable with parameter.

In analogy to the discrete geometric distribution, the continuous exponential distribution determines the waiting time until the first arrival of a Poisson random rare event. The exponential distribution is thus the continuous analogue of the discrete geometric distribution.

Random numbers

Random numbers for the geometric distribution are typically generated using the inversion method. This method is at the geometric distribution particularly, since the individual probabilities of the simple recursion suffice. The inversion method is here so only with rational operations (addition, multiplication ) and without the distribution function to calculate and display previously feasible, which guarantees a fast algorithm for the simulation.