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The memorylessness property asserts that the number of previously failed trials has no effect on the number of future trials needed for a success. Geometric random variables can also be defined as taking values in N 0 {\displaystyle \mathbb {N} _{0}} , which describes the number of failed trials before the first success in a sequence of ...
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.
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.
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 the general model of characterization of probability distribution. Some examples of characterization theorems: The assumption that two linear (or non-linear) statistics are identically distributed (or independent, or have a constancy regression and so on) can be used to characterize various populations. [2]
Likewise, the cumulative distribution of the residual time is = [()]. For large , the distribution is independent of , making it a stationary distribution. An interesting fact is that the limiting distribution of forward recurrence time (or residual time) has the same form as the limiting distribution of the backward recurrence time (or age).
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
Note that the expression of Pinsker inequality depends on what basis of logarithm is used in the definition of KL-divergence. () = .Given the above comments, there is an alternative statement of Pinsker's inequality in some literature that relates information divergence to variation distance: