Editing Geometric distribution (section)

=== Summary statistics ===
The [[mean]] of the geometric distribution is its expected value which is, as previously discussed in [[Geometric distribution#Moments and cumulants|§ Moments and cumulants]], <math>\frac{1}{p}</math> or <math>\frac{1-p}{p}</math> when defined over <math>\mathbb{N}</math> or <math>\mathbb{N}_0</math> respectively.

The [[median]] of the geometric distribution is <math>\left\lceil -\frac{\log 2}{\log(1-p)} \right\rceil</math>when defined over <math>\mathbb{N}</math><ref>{{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}}</ref> and <math>\left\lfloor-\frac{\log 2}{\log(1-p)}\right\rfloor</math> when defined over <math>\mathbb{N}_0</math>.<ref name=":2" />{{Rp|page=69}}

The [[Mode (statistics)|mode]] of the geometric distribution is the first value in the support set. This is 1 when defined over <math>\mathbb{N}</math> and 0 when defined over <math>\mathbb{N}_0</math>.<ref name=":2" />{{Rp|page=69}}

The [[skewness]] of the geometric distribution is <math>\frac{2-p}{\sqrt{1-p}}</math>.<ref name=":0" />{{Rp|pages=|page=115}}

The [[Kurtosis risk|kurtosis]] of the geometric distribution is <math>9 + \frac{p^2}{1-p}</math>.<ref name=":0" />{{Rp|pages=|page=115}} The [[excess kurtosis]] of a distribution is the difference between its kurtosis and the kurtosis of a [[normal distribution]], <math>3</math>.<ref name=":4">{{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}}</ref>{{Rp|pages=|page=217}} Therefore, the excess kurtosis of the geometric distribution is <math>6 + \frac{p^2}{1-p}</math>. Since <math>\frac{p^2}{1-p} \geq 0</math>, the excess kurtosis is always positive so the distribution is [[leptokurtic]].<ref name=":2" />{{Rp|page=69}} In other words, the tail of a geometric distribution decays faster than a Gaussian.<ref name=":4" />{{Rp|pages=|page=217}}