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The only continuous random variable that is memoryless is the exponential random variable. It models random processes like time between consecutive events. [8] The memorylessness property asserts that the amount of time since the previous event has no effect on the future time until the next event occurs.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
The Poisson process is the unique renewal process with the Markov property, [1] as the exponential distribution is the unique continuous random variable with the property of memorylessness. Renewal-reward processes
The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model .
This is a known characteristic of the exponential distribution, i.e., its memoryless property. Intuitively, this means that it does not matter how long it has been since the last renewal epoch, the remaining time is still probabilistically the same as in the beginning of the holding time interval.
For an exponential survival distribution, the probability of failure is the same in every time interval, no matter the age of the individual or device. This fact leads to the "memoryless" property of the exponential survival distribution: the age of a subject has no effect on the probability of failure in the next time interval.
A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made ...
The geometric distribution is the only memoryless discrete probability distribution. [4] It is the discrete version of the same property found in the exponential distribution. [1]: 228 The property asserts that the number of previously failed trials does not affect the number of future trials needed for a success.