Geometric distribution information

Geometric
	Probability mass function
	Cumulative distribution function
Parameters	success probability (real)
Support	k trials where
PMF
CDF	for ,; for
Mean
Median	; (not unique if is an integer)
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF	; for
CF
PGF

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

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

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 $X$ ); 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 $k$ independent trials, each with success probability $p$ . If the probability of success on each trial is $p$ , then the probability that the $k$ -th trial is the first success is

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

for $k=1,2,3,4,\dots$

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:

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

for $k=0,1,2,3,\dots$

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

The geometric distribution is denoted by Geo(p) where $0<p\leq 1$ .^[1]

^ A modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. pp. 48–50, 61–62, 152. ISBN 9781852338961. OCLC 262680588.{{cite book}}: CS1 maint: others (link)

[:0-1] A modern introduction to probability and statistics : understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005. pp. 48–50, 61–62, 152. ISBN 9781852338961. OCLC 262680588.{{cite book}}: CS1 maint: others (link)

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

Geometric distribution information

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