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Probability mass function  
Cumulative distribution function  
Parameters  success probability (real)  success probability (real) 

Support  
Probability mass function (pmf)  
Cumulative distribution function (CDF)  
Mean  
Median  (not unique if is an integer)  (not unique if is an integer) 
Mode  
Variance  
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  , for  
Characteristic function 
Probability mass function  
Cumulative distribution function  
Parameters  success probability (real)  success probability (real) 

Support  
Probability mass function (pmf)  
Cumulative distribution function (CDF)  
Mean  
Median  (not unique if is an integer)  (not unique if is an integer) 
Mode  
Variance  
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  , for  
Characteristic function 
In probability theory and statistics, the geometric distribution is either of two discrete probability distributions:
Which of these one calls "the" geometric distribution is a matter of convention and convenience.
These two different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one (distribution of the number X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by mentioning the support explicitly.
It’s the probability that the first occurrence of success require k number of independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is
for k = 1, 2, 3, ....
The above form of geometric distribution is used for modeling the number of trials until the first success. By contrast, the following form of geometric distribution is used for modeling number of failures until the first success:
for k = 0, 1, 2, 3, ....
In either case, the sequence of probabilities is a geometric sequence.
For example, suppose an ordinary die is thrown repeatedly until the first time a "1" appears. The probability distribution of the number of times it is thrown is supported on the infinite set { 1, 2, 3, ... } and is a geometric distribution with p = 1/6.
The expected value of a geometrically distributed random variable X is 1/p and the variance is (1 − p)/p^{2}:
Similarly, the expected value of the geometrically distributed random variable Y (where Y corresponds to the pmf listed in the right column) is (1 − p)/p, and its variance is (1 − p)/p^{2}:
Let μ = (1 − p)/p be the expected value of Y. Then the cumulants of the probability distribution of Y satisfy the recursion
Outline of proof: That the expected value is (1 − p)/p can be shown in the following way. Let Y be as above. Then
(The interchange of summation and differentiation is justified by the fact that convergent power series converge uniformly on compact subsets of the set of points where they converge.)
For both variants of the geometric distribution, the parameter p can be estimated by equating the expected value with the sample mean. This is the method of moments, which in this case happens to yield maximum likelihood estimates of p.^{[citation needed]}
Specifically, for the first variant let k = k_{1}, ..., k_{n} be a sample where k_{i} ≥ 1 for i = 1, ..., n. Then p can be estimated as
In Bayesian inference, the Beta distribution is the conjugate prior distribution for the parameter p. If this parameter is given a Beta(α, β) prior, then the posterior distribution is^{[citation needed]}
The posterior mean E[p] approaches the maximum likelihood estimate as α and β approach zero.
In the alternative case, let k_{1}, ..., k_{n} be a sample where k_{i} ≥ 0 for i = 1, ..., n. Then p can be estimated as
The posterior distribution of p given a Beta(α, β) prior is^{[citation needed]}
Again the posterior mean E[p] approaches the maximum likelihood estimate as α and β approach zero.
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (March 2011) 
