Memoryless

A variable $x$ is memoryless with respect to $t$ if, for all $s$ with $t\not=0$,

$$P(x > s+t\vert x > t) = P(x > s).$$ (1)

Equivalently,

$${P(x > s+t, x > t)\over P(x > t)} = P(x > s)$$ (2)

$$P(x > s+t) = P(x > s)P(x > t).$$ (3)

The Exponential Distribution, which satisfies

$$P(x > t) = e^{-\lambda t}$$ (4)

$$P(x > s+t) = e^{-\lambda (s+t)},$$ (5)

and therefore

$$P(x > s+t) = P(x > s)P(x > t) = e^{-\lambda s}e^{-\lambda t}$$

$$= e^{-\lambda (s+t)},$$ (6)

is the only memoryless random distribution.

See also Exponential Distribution