Editing Memorylessness (section)

== Characterization of geometric distribution ==
If a discrete probability distribution is memoryless, then it must be the geometric distribution.

From the memorylessness property,<math display="block">\Pr(X>t+s \mid X\geq s)=\Pr(X>t)</math>The definition of [[conditional probability]] reveals that<math display="block">\frac{\Pr(X > t + s)}{\Pr(X \geq s)} = \Pr(X > t)</math>From this it can be proven by induction that <math display="block">\Pr(X > kt) = \Pr(X > 1)^k</math>Then it follows that<math display="block">f_X(x)=Pr(X\leq x)=1-Pr(X>x)=1-Pr(X>1)^x</math> and if we let <math display="block">Pr(X>1)=1-p</math>for some <math>0\leq p \leq 1</math>. we can easily see that X is geometrically distributed with some parameter p. in other words
<math display="block">X\sim Geo(p)</math>