In probability and statistics, memorylessness is a property of probability distributions. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the geometric and exponential distributions are memoryless.

Definition

A random variable is its probability mass function or probability density function is discrete or continuous are nonnegative numbers. In discrete cases, the definition describes the first success in an infinite sequence of independent and identically distributed Bernoulli trials, like the number of coin flips until landing heads. In continuous situations, memorylessness models random phenomena, like the time between two earthquakes. The memorylessness property asserts that the number of previously failed trials or the elapsed time is independent, or has no effect, on the future trials or lead time.

The equality characterizes the geometric and exponential distributions in discrete and continuous contexts respectively. In other words, the geometric random variable is the only discrete memoryless distribution and the exponential random variable is the only continuous memoryless distribution.

so the equality is still satisfied.

Characterization of exponential distribution

If a continuous probability distribution is memoryless, then it must be the exponential distribution.

The definition of conditional probabilityRearranging the equality with the survival functionThis implies that for any natural numberIn general, the equality is true for any rational number. Since the survival function is continuous and rational numbers are dense in the real numbers. This is the survival function of the exponential distribution.

Characterization of geometric distribution

If a discrete probability distribution is memoryless, then it must be the geometric distribution.

The definition of conditional probability

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