Editing Geometric distribution (section)

=== Memorylessness ===
{{Main article|Memorylessness}}
The geometric distribution is the only memoryless discrete probability distribution.<ref>{{Cite book |last1=Dekking |first1=Frederik Michel |url=http://link.springer.com/10.1007/1-84628-168-7 |title=A Modern Introduction to Probability and Statistics |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |publisher=Springer London |isbn=978-1-85233-896-1 |series=Springer Texts in Statistics |location=London |page=50 |language=en |doi=10.1007/1-84628-168-7}}</ref> It is the discrete version of the same property found in the [[exponential distribution]].<ref name=":8">{{Cite book |last1=Johnson |first1=Norman L. |url=https://onlinelibrary.wiley.com/doi/book/10.1002/0471715816 |title=Univariate Discrete Distributions |last2=Kemp |first2=Adrienne W.|author2-link=Adrienne W. Kemp |last3=Kotz |first3=Samuel |date=2005-08-19 |publisher=Wiley |isbn=978-0-471-27246-5 |edition=1 |series=Wiley Series in Probability and Statistics |page= |language=en |doi=10.1002/0471715816}}</ref>{{Rp|page=228}} The property asserts that the number of previously failed trials does not affect the number of future trials needed for a success. 

Because there are two definitions of the geometric distribution, there are also two definitions of memorylessness for discrete random variables.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Memoryless |url=https://mathworld.wolfram.com/ |access-date=2024-07-25 |website=mathworld.wolfram.com |language=en}}</ref> Expressed in terms of [[conditional probability]], the two definitions are<math display="block">\Pr(X>m+n\mid X>n)=\Pr(X>m),</math>

and<math display="block">\Pr(Y>m+n\mid Y\geq n)=\Pr(Y>m),</math>

where <math>m</math> and <math>n</math> are [[Natural number|natural numbers]], <math>X</math> is a geometrically distributed random variable defined over <math>\mathbb{N}</math>, and <math>Y</math> is a geometrically distributed random variable defined over <math>\mathbb{N}_0</math>. Note that these definitions are not equivalent for discrete random variables; <math>Y</math> does not satisfy the first equation and <math>X</math> does not satisfy the second.