What is "memorylessness"

a. (context probability theory English) The property of being memoryless.

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.

As another example, suppose X is the lifetime of a car engine given in terms of number of miles driven. If the engine has lasted 200,000 miles, then, based on our intuition, it is clear that the probability that the engine lasts another 100,000 miles is not the same as the engine lasting 100,000 miles from the first time it was built. However, memorylessness states that the two probabilities are the same (And if our intuition is right, the distribution that describes the lifetime of a large set of these engines does not have the memorylessness property). In essence, we 'forget' what state the car is in. In other words, the probabilities are not influenced by how much time has elapsed.

In the context of Markov processes, memorylessness refers to the Markov property, a different concept which 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. The present article describes the use outside the Markov property, limited to conditional probability distributions.