# Relation between Memorylessness and Independence

I was reading about Memorylessness and Independence and was wondering whether there is any relationship between the two. I realize that memorylessness is a property of a random variable of a particular distribution (such as X is a RV following exponential distribution) and independence has more to do with 2 or more random variables rather than their distribution.

But, is there any fact that links the two?

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Suppose $0<a<b$.

Suppose the probability distribution of $X$ is a memoryless distribution on the interval $(0,\infty)$.

Commonplace mistake: $\Pr(X>b\mid X>a) = \Pr(X>b)$.

Correct statement: $\Pr(X>b\mid X>a) = \Pr(X > b-a)$.

The second thing is what memoryless says. The first thing would be correct if the two events $[X>b]$ and $[X>a]$ were independent. But notice that if $[X>b]$, then necessarily $[X>a]$, so how could they be independent?

So one connection between independence and memorylessness is that memorylessness gets carelessly mistaken for independence in this situation.

But now consider the Poisson process. They waiting time until the next occurrence has a memoryless exponential distribution. This implies that the numbers of occurrences in two time intervals that don't overlap are independent. That's a more substantial connection between memorylessness and independence. Notice what is independent: the number of occurrences in a time interval has a discrete distribution, whose values are in the set $\{0,1,2,3,\ldots\}$. Those numbers of occurrences are what are independent. The things that are memoryless are the distributions of the waiting times. Those have continuous distributions, whose values are in the set $(0,\infty)$.

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The lack of memory or Markov property is a form of conditional independence of RVs.

Consider X$_i$ i=1,2,...t where increasing index may be viewed as moving forward in time. Memoryless property

P[X$_t$=x| X$_t$$_-$$_1$=y$_1$, X$_t$$_-$$_2$=y$_2$, ...X$_1$=y$_t$$_-$$_1$] =

P[X$_t$=x| X$_t$$_-$$_1$$=y$$_1$] for any x and y$_1$,...y$_t$$_-$$_1$

In words this means X$_t$ is INDEPENDENT of X$_t$$_-$$_2$ , ..., X$_1$ CONDITIONAL ON X$_t$$_-$$_1$

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