Memoryless

The exponential distribution often arises, in practice, as being the distribution of the amount of time until some specific event occurs. For instance, the amount of time (starting from now) until an earthquake occurs, or until a new war breaks out, or until a telephone call you receive turns out to be a wrong number are all random variables that tend in practice to have exponential distributions.

We say that a nonnegative random variable X is memoryless if

$P\{X>s+t\Vert X>t\}=P\{X>s\}\qquad \mbox{ for all } s,t\geq 0\qquad (5.1)$
If we think of X as being the lifetime of some instrument, Equation (5.1) states that the probability that the instrument survives for at least s+thours, given that it has survived t hours, is the same as the initial probability alive at age t, the distribution of the remaining amount of time that it survives is the same as the original lifetime distribution (that is, it is as if the instrument does not remember that it has already been in use for a time t)

The condition (5.1) is equivalent to

$\displaystyle\frac{P\{X>s+t,X>t\}}{P\{X>t\}}=P\{X>s\}$or $P\{X>s+t\}=P\{X>s\}+P\{X>t\}\qquad (5.2)$
Since Equation (5.2) is satisfied when X is exponentially distributed (for $e^{-\lambda (s+t)}=e^{-\lambda s}e^{-\lambda t}$), it follows that exponentially distributed random variables are memoryless.


Suppose that X is memoryless and let $\overline{F}(x)=P\{X>x\}$. Then, by Equation (5.2), it follows that

$\overline{F}(s+t)=\overline{F}(s)\overline{F}(t)$
That is, $\overline{F}(\cdot)$ satisfies the functional equation
g(s+t)=g(s)g(t)
However, it turns out that only right continuous solution of this functional equation is
$g(x)=e^{-\lambda x}\qquad (5.3)$
and, since a distribution function is always right continuous, we must have
$\overline{F}(x)=e^{-\lambda x}$ or $F(x)=P\{X\leq x\}=1-e^{\lambda x}$
which shows that X is exponentially distributed.