memoryless random variable

A non-negative-valued random variableMathworldPlanetmath X is memoryless if P(X>s+tX>s)=P(X>t) for s,t0.

In words, given that a certain event did not occur during time period s in the past, the chance that an event will occur after an additional time period t in the future is the same as the chance that the event would occur after a time period t from the beginning, regardless of how long or how short the time period s is; the memory is erased.

From the definition, we see that

P(X>t)=P(X>s+tX>s)=P(X>s+t and X>s)P(X>s)=P(X>s+t)P(X>s),

so P(X>s+t)=P(X>s)P(X>t) iff X is memoryless.

An example of a discrete memoryless random variable is the geometric random variablePlanetmathPlanetmath, since P(X>s+t)=(1-p)s+t=(1-p)s(1-p)t=P(X>s)P(X>t), where p is the probability of X=success. The exponential random variable is an example of a continuous memoryless random variable, which can be proved similarly with 1-p replaced by e-λ. In fact, the exponential random variable is the only continuous random variable having the memoryless property.

Title memoryless random variable
Canonical name MemorylessRandomVariable
Date of creation 2013-03-22 14:39:49
Last modified on 2013-03-22 14:39:49
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 60K05
Classification msc 60G07
Related topic MarkovChain