geometric distribution

The geometric distribution is the discrete probability distribution that describes when the first success in an infinite sequence of independent and identically distributed Bernoulli trials occurs. Its probability mass function depends on its parameterization and support. When supported on $\backslash mathbb\{N\}$, the probability mass function is $$P(X\; =\; k)\; =\; (1\; -\; p)^\{k-1\}\; p$$ where $k\; =\; 1,\; 2,\; 3,\; \backslash dotsc$ is the number of trials and $p$ is the probability of success in each trial.{{Cite book |last1=Nagel |first1=Werner |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9781119243496 |title=Probability and Conditional Expectation: Fundamentals for the Empirical Sciences |last2=Steyer |first2=Rolf |date=2017-04-04 |publisher=Wiley |isbn=978-1-119-24352-6 |edition=1st |series=Wiley Series in Probability and Statistics |pages= |language=en |doi=10.1002/9781119243496}}{{Rp|pages=260–261}}

The support may also be $\backslash mathbb\{N\}\_0$, defining $Y=X-1$. This alters the probability mass function into $$P(Y\; =\; k)\; =\; (1\; -\; p)^k\; p$$ where $k\; =\; 0,\; 1,\; 2,\; \backslash dotsc$ is the number of failures before the first success.{{Cite book |last1=Chattamvelli |first1=Rajan |url=https://link.springer.com/10.1007/978-3-031-02425-2 |title=Discrete Distributions in Engineering and the Applied Sciences |last2=Shanmugam |first2=Ramalingam |publisher=Springer International Publishing |year=2020 |isbn=978-3-031-01297-6 |series=Synthesis Lectures on Mathematics & Statistics |location=Cham |language=en |doi=10.1007/978-3-031-02425-2}}{{Rp|page=66}}

An alternative parameterization of the distribution gives the probability mass function $$P(Y\; =\; k)\; =\; \backslash left(\backslash frac\{P\}\{Q\}\backslash right)^k\; \backslash left(1-\backslash frac\{P\}\{Q\}\backslash right)$$ where $P\; =\; \backslash frac\{1-p\}\{p\}$ and $Q\; =\; \backslash frac\{1\}\{p\}$.{{Rp|pages=208–209}}

An example of a geometric distribution arises from rolling a six-sided die until a "1" appears. Each roll is independent with a $1/6$ chance of success. The number of rolls needed follows a geometric distribution with $p=1/6$.

{{Main article|Memorylessness}}

The geometric distribution is the only memoryless discrete probability distribution.{{Cite book |last1=Dekking |first1=Frederik Michel |url=http://link.springer.com/10.1007/1-84628-168-7 |title=A Modern Introduction to Probability and Statistics |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |publisher=Springer London |isbn=978-1-85233-896-1 |series=Springer Texts in Statistics |location=London |page=50 |language=en |doi=10.1007/1-84628-168-7}} It is the discrete version of the same property found in the exponential distribution.{{Cite book |last1=Johnson |first1=Norman L. |url=https://onlinelibrary.wiley.com/doi/book/10.1002/0471715816 |title=Univariate Discrete Distributions |last2=Kemp |first2=Adrienne W.|author2-link=Adrienne W. Kemp |last3=Kotz |first3=Samuel |date=2005-08-19 |publisher=Wiley |isbn=978-0-471-27246-5 |edition=1 |series=Wiley Series in Probability and Statistics |page= |language=en |doi=10.1002/0471715816}}{{Rp|page=228}} The property asserts that the number of previously failed trials does not affect the number of future trials needed for a success.

Because there are two definitions of the geometric distribution, there are also two definitions of memorylessness for discrete random variables.{{Cite web |last=Weisstein |first=Eric W. |title=Memoryless |url=https://mathworld.wolfram.com/ |access-date=2024-07-25 |website=mathworld.wolfram.com |language=en}} Expressed in terms of conditional probability, the two definitions are$$\backslash Pr(X>m+n\backslash mid\; X>n)=\backslash Pr(X>m),$$

and$$\backslash Pr(Y>m+n\backslash mid\; Y\backslash geq\; n)=\backslash Pr(Y>m),$$

where $m$ and $n$ are natural numbers, $X$ is a geometrically distributed random variable defined over $\backslash mathbb\{N\}$, and $Y$ is a geometrically distributed random variable defined over $\backslash mathbb\{N\}\_0$. Note that these definitions are not equivalent for discrete random variables; $Y$ does not satisfy the first equation and $X$ does not satisfy the second.

The expected value and variance of a geometrically distributed random variable $X$ defined over $\backslash mathbb\{N\}$ is{{Rp|page=261}}$$\backslash operatorname\{E\}(X)\; =\; \backslash frac\{1\}\{p\},$$

\qquad\operatorname{var}(X) = \frac{1-p}{p^2}. With a geometrically distributed random variable $Y$ defined over $\backslash mathbb\{N\}\_0$, the expected value changes into$$\backslash operatorname\{E\}(Y)\; =\; \backslash frac\{1-p\}\; p,$$while the variance stays the same.{{Cite book |last1=Forbes |first1=Catherine |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9780470627242 |title=Statistical Distributions |last2=Evans |first2=Merran |last3=Hastings |first3=Nicholas |last4=Peacock |first4=Brian |date=2010-11-29 |publisher=Wiley |isbn=978-0-470-39063-4 |edition=1st |pages= |language=en |doi=10.1002/9780470627242}}{{Rp|pages=114–115}}

For example, when rolling a six-sided die until landing on a "1", the average number of rolls needed is $\backslash frac\{1\}\{1/6\}\; =\; 6$ and the average number of failures is $\backslash frac\{1\; -\; 1/6\}\{1/6\}\; =\; 5$.

The moment generating function of the geometric distribution when defined over $$

\mathbb{N}

and $\backslash mathbb\{N\}\_0$ respectively is{{Cite book |last1=Bertsekas |first1=Dimitri P. |url=https://archive.org/details/introductiontopr0000bert_p5i9_2ndedi |title=Introduction to probability |last2=Tsitsiklis |first2=John N. |publisher=Athena Scientific |year=2008 |isbn=978-1-886529-23-6 |edition=2nd |series=Optimization and computation series |location=Belmont |page=235 |language=en}}{{Rp|page=114}}$$\backslash begin\{align\}$$

M_X(t) &= \frac{pe^t}{1-(1-p)e^t} \\

M_Y(t) &= \frac{p}{1-(1-p)e^t}, t < -\ln(1-p)

\end{align}The moments for the number of failures before the first success are given by

: $$

\begin{align}

\mathrm{E}(Y^n) & {} =\sum_{k=0}^\infty (1-p)^k p\cdot k^n \\

& {} =p \operatorname{Li}_{-n}(1-p) & (\text{for }n \neq 0)

\end{align}

where $\backslash operatorname\{Li\}\_\{-n\}(1-p)$ is the polylogarithm function.{{Cite web |last=Weisstein |first=Eric W. |title=Geometric Distribution |url=https://mathworld.wolfram.com/ |access-date=2024-07-13 |website=MathWorld |language=en}}

The cumulant generating function of the geometric distribution defined over $\backslash mathbb\{N\}\_0$ is{{Rp|page=216}} $$K(t)\; =\; \backslash ln\; p\; -\; \backslash ln\; (1\; -\; (1-p)e^t)$$The cumulants $\backslash kappa\_r$ satisfy the recursion$$\backslash kappa\_\{r+1\}\; =\; q\; \backslash frac\{\backslash delta\backslash kappa\_r\}\{\backslash delta\; q\},\; r=1,2,\backslash dotsc$$where $q\; =\; 1-p$, when defined over $\backslash mathbb\{N\}\_0$.{{Rp|page=216}}

Consider the expected value $\backslash mathrm\{E\}(X)$ of X as above, i.e. the average number of trials until a success.

The first trial either succeeds with probability $p$, or fails with probability $1-p$.

If it fails, the remaining mean number of trials until a success is identical to the original mean -

this follows from the fact that all trials are independent.

From this we get the formula:

: $\backslash operatorname\; \backslash mathrm\{E\}(X)\; =\; p\; +\; (1-p)(1\; +\; \backslash mathrm\{E\}[X])\; ,$

which, when solved for $\backslash mathrm\{E\}(X)$, gives:

: $\backslash operatorname\; E(X)\; =\; \backslash frac\{1\}\{p\}.$

The expected number of failures $Y$ can be found from the linearity of expectation, $\backslash mathrm\{E\}(Y)\; =\; \backslash mathrm\{E\}(X-1)\; =\; \backslash mathrm\{E\}(X)\; -\; 1\; =\; \backslash frac\; 1\; p\; -\; 1\; =\; \backslash frac\{1-p\}\{p\}$. It can also be shown in the following way:

: $$

\begin{align}

\operatorname E(Y) & =p\sum_{k=0}^\infty(1-p)^k k \\

& = p (1-p) \sum_{k=0}^\infty (1-p)^{k-1} k\\

& = p (1-p) \left(-\sum_{k=0}^\infty \frac{d}{dp}\left[(1-p)^k\right]\right) \\

& = p (1-p) \left[\frac{d}{dp}\left(-\sum_{k=0}^\infty (1-p)^k\right)\right] \\

& = p(1-p)\frac{d}{dp}\left(-\frac{1}{p}\right) \\

& = \frac{1-p}{p}.

\end{align}

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.

The mean of the geometric distribution is its expected value which is, as previously discussed in § Moments and cumulants, $\backslash frac\{1\}\{p\}$ or $\backslash frac\{1-p\}\{p\}$ when defined over $\backslash mathbb\{N\}$ or $\backslash mathbb\{N\}\_0$ respectively.

The median of the geometric distribution is $\backslash left\backslash lceil\; -\backslash frac\{\backslash log\; 2\}\{\backslash log(1-p)\}\; \backslash right\backslash rceil$when defined over $\backslash mathbb\{N\}${{Cite book |last=Aggarwal |first=Charu C. |url=https://link.springer.com/10.1007/978-3-031-53282-5 |title=Probability and Statistics for Machine Learning: A Textbook |publisher=Springer Nature Switzerland |year=2024 |isbn=978-3-031-53281-8 |location=Cham |page=138 |language=en |doi=10.1007/978-3-031-53282-5}} and $\backslash left\backslash lfloor-\backslash frac\{\backslash log\; 2\}\{\backslash log(1-p)\}\backslash right\backslash rfloor$ when defined over $\backslash mathbb\{N\}\_0$.{{Rp|page=69}}

The mode of the geometric distribution is the first value in the support set. This is 1 when defined over $\backslash mathbb\{N\}$ and 0 when defined over $\backslash mathbb\{N\}\_0$.{{Rp|page=69}}

The skewness of the geometric distribution is $\backslash frac\{2-p\}\{\backslash sqrt\{1-p\}\}$.{{Rp|pages=|page=115}}

The kurtosis of the geometric distribution is $9\; +\; \backslash frac\{p^2\}\{1-p\}$.{{Rp|pages=|page=115}} The excess kurtosis of a distribution is the difference between its kurtosis and the kurtosis of a normal distribution, $3$.{{Cite book |last=Chan |first=Stanley |url=https://probability4datascience.com/ |title=Introduction to Probability for Data Science |publisher=Michigan Publishing |year=2021 |isbn=978-1-60785-747-1 |edition=1st |language=en}}{{Rp|pages=|page=217}} Therefore, the excess kurtosis of the geometric distribution is $6\; +\; \backslash frac\{p^2\}\{1-p\}$. Since $\backslash frac\{p^2\}\{1-p\}\; \backslash geq\; 0$, the excess kurtosis is always positive so the distribution is leptokurtic.{{Rp|page=69}} In other words, the tail of a geometric distribution decays faster than a Gaussian.{{Rp|pages=|page=217}}

Entropy is a measure of uncertainty in a probability distribution. For the geometric distribution that models the number of failures before the first success, the probability mass function is:

:$P(X\; =\; k)\; =\; (1\; -\; p)^k\; p,\; \backslash quad\; k\; =\; 0,\; 1,\; 2,\; \backslash dots$

The entropy $H(X)$ for this distribution is defined as:

:$\backslash begin\{align\}$

H(X) &= - \sum_{k=0}^{\infty} P(X = k) \ln P(X = k) \\

&= - \sum_{k=0}^{\infty} (1 - p)^k p \ln \left( (1 - p)^k p \right) \\

&= - \sum_{k=0}^{\infty} (1 - p)^k p \left[ k \ln(1 - p) + \ln p \right] \\

&= -\log p - \frac{1 - p}{p} \log(1 - p)

\end{align}

The entropy increases as the probability $p$ decreases, reflecting greater uncertainty as success becomes rarer.

Fisher information measures the amount of information that an observable random variable $X$ carries about an unknown parameter $p$. For the geometric distribution (failures before the first success), the Fisher information with respect to $p$ is given by:

:$I(p)\; =\; \backslash frac\{1\}\{p^2(1\; -\; p)\}$

Proof:

• The Likelihood Function for a geometric random variable $X$ is:

:$L(p;\; X)\; =\; (1\; -\; p)^X\; p$

• The Log-Likelihood Function is:

:$\backslash ln\; L(p;\; X)\; =\; X\; \backslash ln(1\; -\; p)\; +\; \backslash ln\; p$

• The Score Function (first derivative of the log-likelihood w.r.t. $p$) is:

:$\backslash frac\{\backslash partial\}\{\backslash partial\; p\}\; \backslash ln\; L(p;\; X)\; =\; \backslash frac\{1\}\{p\}\; -\; \backslash frac\{X\}\{1\; -\; p\}$

• The second derivative of the log-likelihood function is:

:$\backslash frac\{\backslash partial^2\}\{\backslash partial\; p^2\}\; \backslash ln\; L(p;\; X)\; =\; -\backslash frac\{1\}\{p^2\}\; -\; \backslash frac\{X\}\{(1\; -\; p)^2\}$

• Fisher Information is calculated as the negative expected value of the second derivative:

:$\backslash begin\{align\}$

I(p) &= -E\left[\frac{\partial^2}{\partial p^2} \ln L(p; X)\right] \\

&= - \left(-\frac{1}{p^2} - \frac{1 - p}{p (1 - p)^2} \right) \\

&= \frac{1}{p^2(1 - p)}

\end{align}

Fisher information increases as $p$ decreases, indicating that rarer successes provide more information about the parameter $p$.

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

:$P(X\; =\; k)\; =\; (1\; -\; p)^\{k\; -\; 1\}\; p,\; \backslash quad\; k\; =\; 1,\; 2,\; 3,\; \backslash dots$

The entropy $H(X)$ for this distribution is given by:

:$\backslash begin\{align\}$

H(X) &= - \sum_{k=1}^{\infty} P(X = k) \ln P(X = k) \\

&= - \sum_{k=1}^{\infty} (1 - p)^{k - 1} p \ln \left( (1 - p)^{k - 1} p \right) \\

&= - \sum_{k=1}^{\infty} (1 - p)^{k - 1} p \left[ (k - 1) \ln(1 - p) + \ln p \right] \\

&= - \log p + \frac{1 - p}{p} \log(1 - p)

\end{align}

Entropy increases as $p$ 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:

:$I(p)\; =\; \backslash frac\{1\}\{p^2(1\; -\; p)\}$

Proof:

• The Likelihood Function for a geometric random variable $X$ is:

:$L(p;\; X)\; =\; (1\; -\; p)^\{X\; -\; 1\}\; p$

• The Log-Likelihood Function is:

:$\backslash ln\; L(p;\; X)\; =\; (X\; -\; 1)\; \backslash ln(1\; -\; p)\; +\; \backslash ln\; p$

• The Score Function (first derivative of the log-likelihood w.r.t. $p$) is:

:$\backslash frac\{\backslash partial\}\{\backslash partial\; p\}\; \backslash ln\; L(p;\; X)\; =\; \backslash frac\{1\}\{p\}\; -\; \backslash frac\{X\; -\; 1\}\{1\; -\; p\}$

• The second derivative of the log-likelihood function is:

:$\backslash frac\{\backslash partial^2\}\{\backslash partial\; p^2\}\; \backslash ln\; L(p;\; X)\; =\; -\backslash frac\{1\}\{p^2\}\; -\; \backslash frac\{X\; -\; 1\}\{(1\; -\; p)^2\}$

• Fisher Information is calculated as the negative expected value of the second derivative:

:$\backslash begin\{align\}$

I(p) &= -E\left[\frac{\partial^2}{\partial p^2} \ln L(p; X)\right] \\

&= - \left(-\frac{1}{p^2} - \frac{1 - p}{p (1 - p)^2} \right) \\

&= \frac{1}{p^2(1 - p)}

\end{align}

• The probability generating functions of geometric random variables $$

X

and $$

Y

defined over $$

\mathbb{N}

and $$

\mathbb{N}_0

are, respectively,{{Rp|pages=114–115}}

::$\backslash begin\{align\}$

G_X(s) & = \frac{s\,p}{1-s\,(1-p)}, \\[10pt]

G_Y(s) & = \frac{p}{1-s\,(1-p)}, \quad |s| < (1-p)^{-1}.

\end{align}

• The characteristic function $\backslash varphi(t)$ is equal to $G(e^\{it\})$ so the geometric distribution's characteristic function, when defined over $$

\mathbb{N}

and $$

\mathbb{N}_0

respectively, is{{Cite book |url=http://link.springer.com/10.1007/978-3-642-04898-2 |title=International Encyclopedia of Statistical Science |publisher=Springer Berlin Heidelberg |year=2011 |isbn=978-3-642-04897-5 |editor-last=Lovric |editor-first=Miodrag |edition=1st |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-642-04898-2}}{{Rp|page=1630}}$$\backslash begin\{align\}$$

\varphi_X(t) &= \frac{pe^{it}}{1-(1-p)e^{it}},\\[10pt]

\varphi_Y(t) &= \frac{p}{1-(1-p)e^{it}}.

\end{align}

• The entropy of a geometric distribution with parameter $p$ is$$-\backslash frac\{p\; \backslash log\_2\; p\; +\; (1-p)\; \backslash log\_2\; (1-p)\}\{p\}$$
• Given a mean, the geometric distribution is the maximum entropy probability distribution of all discrete probability distributions. The corresponding continuous distribution is the exponential distribution.{{Cite journal |last1=Lisman |first1=J. H. C. |last2=Zuylen |first2=M. C. A. van |date=March 1972 |title=Note on the generation of most probable frequency distributions |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1467-9574.1972.tb00152.x |journal=Statistica Neerlandica |language=en |volume=26 |issue=1 |pages=19–23 |doi=10.1111/j.1467-9574.1972.tb00152.x |issn=0039-0402}}
• The geometric distribution defined on $$

\mathbb{N}_0

is infinitely divisible, that is, for any positive integer $n$, there exist $n$ independent identically distributed random variables whose sum is also geometrically distributed. This is because the negative binomial distribution can be derived from a Poisson-stopped sum of logarithmic random variables.{{Rp|pages=606–607}}

• The decimal digits of the geometrically distributed random variable Y are a sequence of independent (and not identically distributed) random variables.{{citation needed|date=May 2012}} For example, the hundreds digit D has this probability distribution:

::$\backslash Pr(D=d)\; =\; \{q^\{100d\}\; \backslash over\; 1\; +\; q^\{100\}\; +\; q^\{200\}\; +\; \backslash cdots\; +\; q^\{900\}\},$

:where q = 1 − p, and similarly for the other digits, and, more generally, similarly for numeral systems with other bases than 10. When the base is 2, this shows that a geometrically distributed random variable can be written as a sum of independent random variables whose probability distributions are indecomposable.

• Golomb coding is the optimal prefix code{{clarify|date=May 2012}} for the geometric discrete distribution.{{Cite journal|last1=Gallager|first1=R.|last2=van Voorhis|first2=D.|date=March 1975|title=Optimal source codes for geometrically distributed integer alphabets (Corresp.)|journal=IEEE Transactions on Information Theory|volume=21|issue=2|pages=228–230|doi=10.1109/TIT.1975.1055357|issn=0018-9448}}

• The sum of $r$ independent geometric random variables with parameter $p$ is a negative binomial random variable with parameters $r$ and $p$.{{Cite book |last=Pitman |first=Jim |url=http://link.springer.com/10.1007/978-1-4612-4374-8 |title=Probability |date=1993 |publisher=Springer New York |isbn=978-0-387-94594-1 |location=New York, NY |page=372 |language=en |doi=10.1007/978-1-4612-4374-8}} The geometric distribution is a special case of the negative binomial distribution, with $r=1$.

• The geometric distribution is a special case of discrete compound Poisson distribution.{{Rp|page=606}}
• The minimum of $n$ geometric random variables with parameters $p\_1,\; \backslash dotsc,\; p\_n$ is also geometrically distributed with parameter $1\; -\; \backslash prod\_\{i=1\}^n\; (1-p\_i)$.{{cite journal |last1=Ciardo |first1=Gianfranco |last2=Leemis |first2=Lawrence M. |last3=Nicol |first3=David |date=1 June 1995 |title=On the minimum of independent geometrically distributed random variables |url=https://dx.doi.org/10.1016/0167-7152%2894%2900130-Z |journal=Statistics & Probability Letters |language=en |volume=23 |issue=4 |pages=313–326 |doi=10.1016/0167-7152(94)00130-Z |s2cid=1505801 |hdl-access=free |hdl=2060/19940028569}}

• Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable X_k has a Poisson distribution with expected value r^k/k. Then

::$\backslash sum\_\{k=1\}^\backslash infty\; k\backslash ,X\_k$

:has a geometric distribution taking values in $\backslash mathbb\{N\}\_0$, with expected value r/(1 − r).{{citation needed|date=May 2012}}

• The exponential distribution is the continuous analogue of the geometric distribution. Applying the floor function to the exponential distribution with parameter $\backslash lambda$ creates a geometric distribution with parameter $p=1-e^\{-\backslash lambda\}$ defined over $\backslash mathbb\{N\}\_0$.{{Rp|page=74}} This can be used to generate geometrically distributed random numbers as detailed in § Random variate generation.

• If p = 1/n and X is geometrically distributed with parameter p, then the distribution of X/n approaches an exponential distribution with expected value 1 as n → ∞, since$$$$

\begin{align}

\Pr(X/n>a)=\Pr(X>na) & = (1-p)^{na} = \left(1-\frac 1 n \right)^{na} = \left[ \left( 1-\frac 1 n \right)^n \right]^{a} \\

& \to [e^{-1}]^{a} = e^{-a} \text{ as } n\to\infty.

\end{align}

More generally, if p = λ/n, where λ is a parameter, then as n→ ∞ the distribution of X/n approaches an exponential distribution with rate λ:$\backslash Pr(X>nx)=\backslash lim\_\{n\; \backslash to\; \backslash infty\}(1-\backslash lambda\; /n)^\{nx\}=e^\{-\backslash lambda\; x\}$ therefore the distribution function of X/n converges to $1-e^\{-\backslash lambda\; x\}$, which is that of an exponential random variable.{{Cn|date=July 2024}}

• The index of dispersion of the geometric distribution is $\backslash frac\{1\}\{p\}$ and its coefficient of variation is $\backslash frac\{1\}\{\backslash sqrt\{1-p\}\}$. The distribution is overdispersed.{{Rp|page=216}}

The true parameter $p$ of an unknown geometric distribution can be inferred through estimators and conjugate distributions.

Provided they exist, the first $l$ moments of a probability distribution can be estimated from a sample $x\_1,\; \backslash dotsc,\; x\_n$ using the formula$$m\_i\; =\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{j=1\}^n\; x^i\_j$$where $m\_i$ is the $i$th sample moment and $1\; \backslash leq\; i\; \backslash leq\; l$.{{Cite book |last1=Evans |first1=Michael |url=https://www.utstat.toronto.edu/mikevans/jeffrosenthal/ |title=Probability and Statistics: The Science of Uncertainty |last2=Rosenthal |first2=Jeffrey |year=2023 |isbn=978-1429224628 |edition=2nd |pages= |publisher=Macmillan Learning |language=en}}{{Rp|pages=349–350}} Estimating $\backslash mathrm\{E\}(X)$ with $m\_1$ gives the sample mean, denoted $$

\bar{x}

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

p

gives the estimators $$

\hat{p} = \frac{1}{\bar{x}}

and $$

\hat{p} = \frac{1}{\bar{x}+1}

when supported on $\backslash mathbb\{N\}$ and $\backslash mathbb\{N\}\_0$ respectively. These estimators are biased since $\backslash mathrm\{E\}\backslash left(\backslash frac\{1\}\{\backslash bar\{x\}\}\backslash right)\; >\; \backslash frac\{1\}\{\backslash mathrm\{E\}(\backslash bar\{x\})\}\; =\; p$ as a result of Jensen's inequality.{{Cite book |last1=Held |first1=Leonhard |url=https://link.springer.com/10.1007/978-3-662-60792-3 |title=Likelihood and Bayesian Inference: With Applications in Biology and Medicine |last2=Sabanés Bové |first2=Daniel |date=2020 |publisher=Springer Berlin Heidelberg |isbn=978-3-662-60791-6 |series=Statistics for Biology and Health |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-662-60792-3}}{{Rp|pages=53–54}}

The maximum likelihood estimator of $p$ is the value that maximizes the likelihood function given a sample.{{Rp|page=308}} By finding the zero of the derivative of the log-likelihood function when the distribution is defined over $\backslash mathbb\{N\}$, the maximum likelihood estimator can be found to be $\backslash hat\{p\}\; =\; \backslash frac\{1\}\{\backslash bar\{x\}\}$, where $\backslash bar\{x\}$ is the sample mean.{{Cite web |last=Siegrist |first=Kyle |date=2020-05-05 |title=7.3: Maximum Likelihood |url=https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.03%3A_Maximum_Likelihood |access-date=2024-06-20 |website=Statistics LibreTexts |language=en}} If the domain is $\backslash mathbb\{N\}\_0$, then the estimator shifts to $\backslash hat\{p\}\; =\; \backslash frac\{1\}\{\backslash bar\{x\}+1\}$. As previously discussed in § Method of moments, these estimators are biased.

Regardless of the domain, the bias is equal to

: $$

b \equiv \operatorname{E}\bigg[\;(\hat p_\mathrm{mle} - p)\;\bigg]

= \frac{p\,(1-p)}{n}

which yields the bias-corrected maximum likelihood estimator,{{Cn|date=July 2024}}

: $$

\hat{p\,}^*_\text{mle} = \hat{p\,}_\text{mle} - \hat{b\,}

In Bayesian inference, the parameter $p$ is a random variable from a prior distribution with a posterior distribution calculated using Bayes' theorem after observing samples.{{Rp|page=167}} If a beta distribution is chosen as the prior distribution, then the posterior will also be a beta distribution and it is called the conjugate distribution. In particular, if a $\backslash mathrm\{Beta\}(\backslash alpha,\backslash beta)$ prior is selected, then the posterior, after observing samples $k\_1,\; \backslash dotsc,\; k\_n\; \backslash in\; \backslash mathbb\{N\}$, is{{Cite CiteSeerX |citeseerx=10.1.1.157.5540 |first=Daniel |last=Fink |title=A Compendium of Conjugate Priors}}$$p\; \backslash sim\; \backslash mathrm\{Beta\}\backslash left(\backslash alpha+n,\backslash \; \backslash beta+\backslash sum\_\{i=1\}^n\; (k\_i-1)\backslash right).\; \backslash !$$Alternatively, if the samples are in $\backslash mathbb\{N\}\_0$, the posterior distribution is{{Cite web|url=http://halweb.uc3m.es/esp/Personal/personas/mwiper/docencia/English/PhD_Bayesian_Statistics/ch3_2009.pdf |archive-url=https://web.archive.org/web/20100408092905/http://halweb.uc3m.es/esp/Personal/personas/mwiper/docencia/English/PhD_Bayesian_Statistics/ch3_2009.pdf |archive-date=2010-04-08 |url-status=live|title=3. Conjugate families of distributions}}$$p\; \backslash sim\; \backslash mathrm\{Beta\}\backslash left(\backslash alpha+n,\backslash beta+\backslash sum\_\{i=1\}^n\; k\_i\backslash right).$$Since the expected value of a $\backslash mathrm\{Beta\}(\backslash alpha,\backslash beta)$ distribution is $\backslash frac\{\backslash alpha\}\{\backslash alpha+\backslash beta\}$,{{Rp|page=145}} as $\backslash alpha$ and $\backslash beta$ approach zero, the posterior mean approaches its maximum likelihood estimate.

{{Further|Non-uniform random variate generation}}

The geometric distribution can be generated experimentally from i.i.d. standard uniform random variables by finding the first such random variable to be less than or equal to $p$. However, the number of random variables needed is also geometrically distributed and the algorithm slows as $p$ decreases.{{Cite book |last=Devroye |first=Luc |url=http://link.springer.com/10.1007/978-1-4613-8643-8 |title=Non-Uniform Random Variate Generation |publisher=Springer New York |year=1986 |isbn=978-1-4613-8645-2 |location=New York, NY |language=en |doi=10.1007/978-1-4613-8643-8}}{{Rp|page=498}}

Random generation can be done in constant time by truncating exponential random numbers. An exponential random variable $E$ can become geometrically distributed with parameter $p$ through $\backslash lceil\; -E/\backslash log(1-p)\; \backslash rceil$. In turn, $E$ can be generated from a standard uniform random variable $U$ altering the formula into $\backslash lceil\; \backslash log(U)\; /\; \backslash log(1-p)\backslash rceil$.{{Rp|pages=499–500}}{{Cite book |last=Knuth |first=Donald Ervin |title=The Art of Computer Programming |publisher=Addison-Wesley |year=1997 |isbn=978-0-201-89683-1 |edition=3rd |volume=2 |location=Reading, Mass |pages=136 |language=en}}

The geometric distribution is used in many disciplines. In queueing theory, the M/M/1 queue has a steady state following a geometric distribution.{{Cite book |last=Daskin |first=Mark S. |url=https://link.springer.com/10.1007/978-3-031-02493-1 |title=Bite-Sized Operations Management |publisher=Springer International Publishing |year=2021 |isbn=978-3-031-01365-2 |series=Synthesis Lectures on Operations Research and Applications |location=Cham |page=127 |language=en |doi=10.1007/978-3-031-02493-1}} In stochastic processes, the Yule Furry process is geometrically distributed.{{Cite book |last1=Madhira |first1=Sivaprasad |url=https://link.springer.com/10.1007/978-981-99-5601-2 |title=Introduction to Stochastic Processes Using R |last2=Deshmukh |first2=Shailaja |publisher=Springer Nature Singapore |year=2023 |isbn=978-981-99-5600-5 |location=Singapore |page=449 |language=en |doi=10.1007/978-981-99-5601-2}} The distribution also arises when modeling the lifetime of a device in discrete contexts.{{Citation |last1=Gupta |first1=Rakesh |title=Some Discrete Parametric Markov–Chain System Models to Analyze Reliability |date=2023 |work=Advances in Reliability, Failure and Risk Analysis |pages=305–306 |editor-last=Garg |editor-first=Harish |url=https://link.springer.com/10.1007/978-981-19-9909-3_14 |access-date=2024-07-13 |place=Singapore |publisher=Springer Nature Singapore |language=en |doi=10.1007/978-981-19-9909-3_14 |isbn=978-981-19-9908-6 |last2=Gupta |first2=Shubham |last3=Ali |first3=Irfan}} It has also been used to fit data including modeling patients spreading COVID-19.{{Cite journal |last=Polymenis |first=Athanase |date=2021-10-01 |title=An application of the geometric distribution for assessing the risk of infection with SARS-CoV-2 by location |url=https://www.nepjol.info/index.php/AJMS/article/view/38783 |journal=Asian Journal of Medical Sciences |volume=12 |issue=10 |pages=8–11 |doi=10.3126/ajms.v12i10.38783 |issn=2091-0576|doi-access=free }}

• Hypergeometric distribution
• Coupon collector's problem
• Compound Poisson distribution
• Negative binomial distribution

{{reflist}}

{{ProbDistributions|discrete-infinite}}

Category:Discrete distributions

Category:Exponential family distributions

Category:Infinitely divisible probability distributions

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