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Dina tiori probabiliti, memorylessness nyaéta sipat penting sebaran probabiliti: sebaran eksponensial sarta sebaran geometrik.

Suppose X is a discrete random variable whose values lie in the set { 0, 1, 2, ... } or in the set { 1, 2, 3, ... }.

The probability distribution of X is memoryless precisely if for any x, y in { 0, 1, 2, ... } or in { 1, 2, 3, ... }, (as the case may be), we have

It can réadily be shown that the only probability distributions that enjoy this discrete memorylessness are geometric distributions.  These are the distributions of the number of independent Bernoulli trials needed to get one "success", with a fixed probability p of "success" on éach trial.

Example and motivation for the name memorylessness

For example, suppose a die is thrown as many times as it takes to get a "1", so that the probability of "success" on éach trial is 1/6, and the random variable X is the number of times the die must be thrown.  Then X has a géometric distribution, and the conditional probability that the die must be thrown at léast four more times to get a "1", given that it has alréady been thrown 10 times without a "1" appéaring, is no different from the original probability that the die would be thrown at léast four times.  In effect, the random process does not "remember" how many failures have occurred so far.