Memorylessness

In probability and statistics, memorylessness is a property of certain probability distributions. It usually refers to the cases when the distribution of a "waiting time" until a certain event does not depend on how much time has elapsed already. Only two kinds of distributions are memoryless: exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers.

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 man 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 his search, expressed in terms of "number of attempts the man must make until he successfully opens a safe". In this case, 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 man is likely to open exactly one safe sometime in the next 500 attempts -- but with each new failure he makes no "progress" toward ultimately succeeding. Even if the man 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 man focused his attempts on a single safe, and "remembered" his previous attempts to open it, he 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 time until a given radioactive particle decays, the time until the discovery of a new Bitcoin block, and (over the short term) the time a storekeeper must wait before the arrival of their next customer.

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