); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.

The geometric distribution gives the probability that the first occurrence of success requires

The above form of the geometric distribution is used for modeling the number of trials up to and including the first success.

By contrast, the following form of the geometric distribution is used for modeling the number of failures until the first success: for

Its probability mass function depends on its parameterization and support.

[3]: 66 An alternative parameterization of the distribution gives the probability mass function

[1]: 208–209 An example of a geometric distribution arises from rolling a six-sided die until a "1" appears.

The number of rolls needed follows a geometric distribution with

[4] It is the discrete version of the same property found in the exponential distribution.

Note that these definitions are not equivalent for discrete random variables;

The expected value and variance of a geometrically distributed random variable

The moment generating function of the geometric distribution when defined over

[8] The cumulant generating function of the geometric distribution defined over

If we fail the remaining mean number of trials until a success is identical to the original mean.

The mean of the geometric distribution is its expected value which is, as previously discussed in § Moments and cumulants,

[3]: 69  In other words, the tail of a geometric distribution decays faster than a Gaussian.

For the geometric distribution that models the number of failures before the first success, the probability mass function is: The entropy

decreases, reflecting greater uncertainty as success becomes rarer.

For the geometric distribution (failures before the first success), the Fisher information with respect to

decreases, indicating that rarer successes provide more information about the parameter

For the geometric distribution modeling the number of trials until the first success, the probability mass function is: The entropy

decreases, reflecting greater uncertainty as the probability of success in each trial becomes smaller.

Fisher information for the geometric distribution modeling the number of trials until the first success is given by: Proof: The true parameter

Substituting this estimate in the formula for the expected value of a geometric distribution and solving for

[16]: 308  By finding the zero of the derivative of the log-likelihood function when the distribution is defined over

As previously discussed in § Method of moments, these estimators are biased.

Regardless of the domain, the bias is equal to which yields the bias-corrected maximum likelihood estimator,[citation needed] In Bayesian inference, the parameter

However, the number of random variables needed is also geometrically distributed and the algorithm slows as

[24] The distribution also arises when modeling the lifetime of a device in discrete contexts.

[25] It has also been used to fit data including modeling patients spreading COVID-19.