Memorylessness

For the use of the term in materials science, see hysteresis.

In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers.

The property is most easily explained in terms of "waiting times". Suppose that a random variable, X, is defined to be the time elapsed in a shop from 9 am on a certain day until the arrival of the first customer: thus X is the time a server waits for the first customer. The "memoryless" property makes a comparison between the probability distributions of the time a server has to wait from 9 am onwards for his first customer, and the time that the server still has to wait for the first customer on those occasions when no customer has arrived by any given later time: the property of memorylessness is that these distributions of "time from now to the next customer" are exactly the same.

The terms "memoryless" and "memorylessness" have sometimes been used in a slightly different way to refer to Markov processes in which the underlying assumption of the Markov property implies that the properties of random variables related to the future depend only on relevant information about the current time, not on information from further in the past. While these different meanings of memorylessness are connected at a deeply theoretical level,^[1] the present article describes its use in relation to probability distributions.

Discrete memorylessness

Suppose X is a discrete random variable whose values lie in the set { 0, 1, 2, ... }. The probability distribution of X is memoryless precisely if for any m, n in { 0, 1, 2, ... }, we have

$\Pr(X>m+n \mid X \geq m)=\Pr(X>n).$

Here, Pr(X > m + n | X  ≥  m) denotes the conditional probability that the value of X is larger than m + n, given that it is larger than or equal to m.

The only memoryless discrete probability distributions are the geometric distributions, which feature the number of independent Bernoulli trials needed to get one "success", with a fixed probability p of "success" on each trial. In other words those are the distributions of waiting time in a Bernoulli process.

A frequent misunderstanding

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

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

It does not mean that

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

which would be true only if the events X > 40 and X ≥ 30 were independent, which cannot be the case.

Continuous memorylessness

Suppose X is a continuous random variable whose values lie in the non-negative real numbers [0, ∞). The probability distribution of X is memoryless precisely if for any non-negative real numbers t and s, we have

$\Pr(X>t+s \mid X>t)=\Pr(X>s).\,$

This is similar to the discrete version except that s and t are constrained only to be non-negative real numbers instead of integers. Rather than counting trials until the first "success", for example, we may be marking time until the arrival of the first phone call at a switchboard.

Geometric distributions and exponential distributions are discrete and continuous analogs.

The memoryless distributions are the exponential distributions

The only memoryless continuous probability distributions are the exponential distributions, so memorylessness completely characterizes the exponential distributions among all continuous ones.

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

$G(t) = \Pr(X > t).\,$

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

$\Pr(X > t + s | X > t) = \Pr(X > s)\,$

and the definition of conditional probability, it follows that

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

This gives the functional equation

$G(t + s) = G(t) G(s)\,$

and solutions of this can be sought under the condition that G is a monotone decreasing function.

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

Notes

1. ^ Feller (1971)

References

• Feller, W. (1971) Introduction to Probability Theory and Its Applications, Vol II (2nd edition),Wiley. Section I.3 ISBN 0-471-25709-5