Geometric distribution

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 ${\displaystyle k\in \{0,1,2,3,\dots \}\!}$
Probability mass function (pmf)	${\displaystyle (1-p)^{k-1}\,p\!}$	${\displaystyle (1-p)^{k}\,p\!}$
CDF	${\displaystyle 1-(1-p)^{k}\!}$	${\displaystyle 1-(1-p)^{k+1}\!}$
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	${\displaystyle {\frac {2-p}{\sqrt {1-p}}}\!}$	${\displaystyle {\frac {2-p}{\sqrt {1-p}}}\!}$
Excess kurtosis	${\displaystyle 6+{\frac {p^{2}}{1-p}}\!}$	${\displaystyle 6+{\frac {p^{2}}{1-p}}\!}$
Entropy	${\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}\!}$	${\displaystyle {\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}\!}$
MGF	${\displaystyle {\frac {pe^{t}}{1-(1-p)e^{t}}}\!}$, for ${\displaystyle t<-\ln(1-p)\!}$	${\displaystyle {\frac {p}{1-(1-p)e^{t}}}\!}$
CF	${\displaystyle {\frac {pe^{it}}{1-(1-p)\,e^{it}}}\!}$	${\displaystyle {\frac {p}{1-(1-p)\,e^{it}}}\!}$

(not unique if ${\displaystyle -1/\log _{2}(1-p)}$ is an integer)

(not unique if ${\displaystyle -1/\log _{2}(1-p)}$ is an integer)

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 requires 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

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Wikipedia