Geometric distribution

In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:

• The probability distribution of the number ${\displaystyle X}$ of Bernoulli trials needed to get one success, supported on the set ${\displaystyle \{1,2,3,\ldots \}}$;
• The probability distribution of the number ${\displaystyle Y=X-1}$ of failures before the first success, supported on the set ${\displaystyle \{0,1,2,\ldots \}}$.

Geometric

Probability mass function
Cumulative distribution function
Parameters	${\displaystyle 0<p\leq 1}$ success probability (real)	${\displaystyle 0<p\leq 1}$ success probability (real)
Support	k trials where ${\displaystyle k\in \{1,2,3,\dots \}}$	k failures where ${\displaystyle k\in \{0,1,2,3,\dots \}}$
PMF	${\displaystyle (1-p)^{k-1}p}$	${\displaystyle (1-p)^{k}p}$
CDF	${\displaystyle 1-(1-p)^{\lfloor x\rfloor }}$ for ${\displaystyle x\geq 1}$, ${\displaystyle 0}$ for ${\displaystyle x<1}$	${\displaystyle 1-(1-p)^{\lfloor x\rfloor +1}}$ for ${\displaystyle x\geq 0}$, ${\displaystyle 0}$ for ${\displaystyle x<0}$
Mean	${\displaystyle {\frac {1}{p}}}$	${\displaystyle {\frac {1-p}{p}}}$
Median	${\displaystyle \left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil }$ (not unique if ${\displaystyle -1/\log _{2}(1-p)}$ is an integer)	${\displaystyle \left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil -1}$ (not unique if ${\displaystyle -1/\log _{2}(1-p)}$ is an integer)
Mode	${\displaystyle 1}$	${\displaystyle 0}$
Variance	${\displaystyle {\frac {1-p}{p^{2}}}}$	${\displaystyle {\frac {1-p}{p^{2}}}}$
Skewness	${\displaystyle {\frac {2-p}{\sqrt {1-p}}}}$	${\displaystyle {\frac {2-p}{\sqrt {1-p}}}}$
Ex. kurtosis	${\displaystyle 6+{\frac {p^{2}}{1-p}}}$	${\displaystyle 6+{\frac {p^{2}}{1-p}}}$
Entropy	${\displaystyle {\tfrac {-(1-p)\log(1-p)-p\log p}{p}}}$	${\displaystyle {\tfrac {-(1-p)\log(1-p)-p\log p}{p}}}$
MGF	${\displaystyle {\frac {pe^{t}}{1-(1-p)e^{t}}},}$ for ${\displaystyle t<-\ln(1-p)}$	${\displaystyle {\frac {p}{1-(1-p)e^{t}}},}$ for ${\displaystyle t<-\ln(1-p)}$
CF	${\displaystyle {\frac {pe^{it}}{1-(1-p)e^{it}}}}$	${\displaystyle {\frac {p}{1-(1-p)e^{it}}}}$
PGF	${\displaystyle {\frac {pz}{1-(1-p)z}}}$	${\displaystyle {\frac {p}{1-(1-p)z}}}$

Which of these is called 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 ${\displaystyle X}$); 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 ${\displaystyle k}$ independent trials, each with success probability ${\displaystyle p}$. If the probability of success on each trial is ${\displaystyle p}$, then the probability that the ${\displaystyle k}$-th trial is the first success is

${\displaystyle \Pr(X=k)=(1-p)^{k-1}p}$

for ${\displaystyle k=1,2,3,4,\dots }$

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:

${\displaystyle \Pr(Y=k)=\Pr(X=k+1)=(1-p)^{k}p}$

for ${\displaystyle k=0,1,2,3,\dots }$

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 ${\displaystyle \{1,2,3,\dots \}}$ and is a geometric distribution with ${\displaystyle p=1/6}$.

The geometric distribution is denoted by Geo(p) where ${\displaystyle 0<p\leq 1}$.