Geometric Distribution (Geom₍p₎) ○◂｜appendage｜___｜20260628174208-01-•

Geometric	…	…	…
Probability mass function	…	…	…
Cumulative distribution function	…	…	…
Notation	$\mathrm{Geom}(p)$	…	…
Parameters	$0<p\le 1$ success probability (real)	$0<p\le 1$ success probability (real)
Support	k trials where $k\in \mathbb{N}=\{1,2,3,\dots \}$	k failures where $k\in {\mathbb{N}}_0=\{0,1,2,\dots \}$
PMF	$(1-p{)}^{k-1}p$	$(1-p{)}^{k}p$
CDF	$1-(1-p{)}^{\lfloor x\rfloor }$ for $x\ge 1$, $0$ for $x<1$	$1-(1-p{)}^{\lfloor x\rfloor +1}$ for $x\ge 0$, $0$ for $x<0$
Mean	$\frac{1}{p}$	$\frac{1-p}{p}$
Median	$\lceil \frac{-1}{{\log }_2(1-p)}\rceil$ (not unique if $-1/{\log }_2(1-p)$ is an integer)	$\lceil \frac{-1}{{\log }_2(1-p)}\rceil -1$ (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	$\frac{-(1-p)\log (1-p)-p\log p}{p}$	$\frac{-(1-p)\log (1-p)-p\log p}{p}$
MGF	$\frac{p{e}^t}{1-(1-p)e^t},$ for $t<-\ln (1-p)$	$\frac{p}{1-(1-p)e^t},$ for $t<-\ln (1-p)$
CF	$\frac{p{e}^{it}}{1-(1-p)e^{it}}$	$\frac{p}{1-(1-p)e^{it}}$
PGF	$\frac{pz}{1-(1-p)z}$	$\frac{p}{1-(1-p)z}$
Fisher information	$\frac{1}{p^2(1-p)}$	$\frac{1}{p^2(1-p)}$

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