quality:correction
Memorylessness

The geometric distribution is the only memoryless discrete probability distribution. It is the discrete version of the same property found in the exponential distribution. 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. Expressed in terms of conditional probability, the two definitions are Pr (X > m + n ∣ X > n) = Pr (X > m), {\displaystyle \Pr(X>m+n\mid X>n)=\Pr(X>m),}

and Pr (Y > m + n ∣ Y ≥ n) = Pr (Y > m), {\displaystyle \Pr(Y>m+n\mid Y\geq n)=\Pr(Y>m),}

where m {\displaystyle m} and n {\displaystyle n} are natural numbers, X {\displaystyle X} is a geometrically distributed random variable defined over N {\displaystyle \mathbb {N} }, and Y {\displaystyle Y} is a geometrically distributed random variable defined over N 0 {\displaystyle \mathbb {N} _{0}}. Note that these definitions are not equivalent for discrete random variables; Y {\displaystyle Y} does not satisfy the first equation and X {\displaystyle X} does not satisfy the second.