Memorylessness

Most phenomena are not memoryless, which means that observers will obtain information about them over time. For example, suppose that X is a random variable, the lifetime of a car engine, expressed in terms of "number of miles driven until the engine breaks down". It is clear, based on our intuition, that an engine which has already been driven for 300,000 miles will have a much lower X than would a second (equivalent) engine which has only been driven for 1,000 miles. Hence, this random variable would not have the memorylessness property.

In contrast, let us examine a situation which would exhibit memorylessness. Imagine a long hallway, lined on one wall with thousands of safes. Each safe has a dial with 500 positions, and each has been assigned an opening position at random. Imagine that an eccentric person walks down the hallway, stopping once at each safe to make a single random attempt to open it. In this case, we might define random variable X as the lifetime of their search, expressed in terms of "number of attempts the person must make until they successfully open a safe". In this case, E[X] will always be equal to the value of 500, regardless of how many attempts have already been made. Each new attempt has a (1/500) chance of succeeding, so the person is likely to open exactly one safe sometime in the next 500 attempts – but with each new failure they make no "progress" toward ultimately succeeding. Even if the safe-cracker has just failed 499 consecutive times (or 4,999 times), we expect to wait 500 more attempts until we observe the next success. If, instead, this person focused their attempts on a single safe, and "remembered" their previous attempts to open it, they would be guaranteed to open the safe after, at most, 500 attempts (and, in fact, at onset would only expect to need 250 attempts, not 500).

Real-life examples of memorylessness include the universal law of radioactive decay, which describes the time until a given radioactive particle decays, and the time until the discovery of a new Bitcoin block. An often used (theoretical) example of memorylessness in queueing theory is the time a storekeeper must wait before the arrival of the next customer.

"Memorylessness" of the probability distribution of the number of trials X until the first success means that, for example,

${\displaystyle \Pr(X>40\mid X\geq 30)=\Pr(X>10).}$

It does not mean that

${\displaystyle \Pr(X>40\mid X\geq 30)=\Pr(X>40),}$

which would be true only if the events X > 40 and X ≥ 30 were independent, i.e. ${\displaystyle \Pr(X\geq 30)=1.}$

The only memoryless continuous probability distribution is the exponential distribution, so memorylessness completely characterizes the exponential distribution among all continuous ones. The property is derived through the following proof:

To see this, first define the survival function, S, as

${\displaystyle S(t)=\Pr(X>t).}$

Note that S(t) is then monotonically decreasing. From the relation

${\displaystyle \Pr(X>t+s\mid X>t)=\Pr(X>s)}$

and the definition of conditional probability, it follows that

${\displaystyle {\frac {\Pr(X>t+s)}{\Pr(X>t)}}=\Pr(X>s).}$

This gives the functional equation (which is a result of the memorylessness property):

${\displaystyle S(t+s)=S(t)S(s)}$

From this, we must have for example:

${\displaystyle S(2)=S(1)^{2}\quad }$

${\displaystyle S(1)=S(1/2)^{2}{\text{ i.e.}}\quad S(1/2)=S(1)^{1/2}.}$

In general:

${\displaystyle S(a)=S(1)^{a}}$

The only continuous function that will satisfy this equation for any positive, rational a is:

${\displaystyle S(a)=S(1)^{a}=e^{\ln(S(1))a}=e^{-\lambda a},}$

where ${\displaystyle \lambda =-\ln(S(1)).}$

Therefore, since S(a) is a probability and must have ${\displaystyle \lambda >0,}$ then any memorylessness function must be an exponential.

Put a different way, S is a monotone decreasing function (meaning that for times ${\displaystyle x\leq y,}$ then ${\displaystyle S(x)\geq S(y).}$)

The functional equation alone will imply that S restricted to rational multiples of any particular number is an exponential function. Combined with the fact that S is monotone, this implies that S over its whole domain is an exponential function.