memoryless random variable

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

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

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

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.