# memoryless random variable

A non-negative-valued random variable $X$ is memoryless if $P(X>s+t\mid X>s)=P(X>t)$ for $s,t\geq 0$.

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

 $P(X>t)=P(X>s+t\mid X>s)=\frac{P(X>s+t\mbox{ and }X>s)}{P(X>s)}=\frac{P(X>s+t)}% {P(X>s)},$

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 variable, 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 $e^{-\lambda}$. In fact, the exponential random variable is the only continuous random variable having the memoryless property.

Title memoryless random variable MemorylessRandomVariable 2013-03-22 14:39:49 2013-03-22 14:39:49 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 60K05 msc 60G07 MarkovChain