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The purpose of this question is to gather material about "lack of memory" and to add new ideas about that.

If the conditional distribution of a given distribution is equal to unconditional distribution then that distribution possesses a "lack of memory" property. The exponential distribution contains "lack of memory". So my questions are

1) Is there any other distribution which has "lack of memory"?

2) What special properties are associated with exponential or other distributions (if exist) because of "lack of memory"?

Please provide interesting examples on applied side which will be helpful for learners to understand lack of memory. One example is here.

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    $\begingroup$ For Q1, the wikipedia page you link shows that by the definition of Memorylessness ($P( T>s+t | T>s ) = P(T>t)$), one can derive the only cont. distribution with the property is the exponential. I believe that if you assume a discrete distribution, the only solution is geometric. $\endgroup$ Jul 28, 2013 at 15:01
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    $\begingroup$ @Cam.Davidson.Pilon yes, Geometric distribution posses "lack of memory", solution is here $\endgroup$
    – SAAN
    Jul 28, 2013 at 15:10
  • $\begingroup$ I'm more interested in the if and only if part, and yes, the wikipedia article you linked suggests that. $\endgroup$ Jul 28, 2013 at 15:33

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You have misstated the condition of memorylessness in your question. A random variable $T \geqslant 0$ is said to have the memoryless property if it has the same distribution as $T-s|T>s$ for all $s \geqslant 0$. If $T$ is interpreted as the time until some event then we can interpret memorylessness as saying that the distribution of the remaining time to the event is not affected by the elapsed time.

It is easily shown that the exponential distribution is the only continuous distribution with this property, and the geometric distribution is the only discrete (integer) distribution with this property. Since memorylessness implies a particular distributional family in either case, formally speaking, every property of those families is a logical implication of memorylessness.

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