Talk:Memorylessness

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How to prove that exponential distribution is the ONLY distribution that has the memoriless property?

That is an excellent question, and at some point soon I'll add something to the article on this point. Here's the quick version of the answer:

Let G(t) = Pr(X > t).

Then basic laws of probability quickly imply that G(t) gets smaller as t gets bigger. The memorylessness of this distribution is expressed as

Pr(X > t + s | X > t) = Pr(X > s).

By the definition of conditional probability, this implies

Pr(X > t + s)/Pr(X > t) = Pr(X > s).

Thus we have the functional equation

G(t + s) = G(t) G(s)

AND we have the fact that G is a monotone decreasing function.

The functional equation alone will imply that G restricted to rational multiples of any particular number is an exponential function. Combined with the fact that G is monotone, this implies G on its whole domain is an exponential function.

That's a bit quick and hand-waving, but the detailed proof can be reconstructed from it. Michael Hardy 00:00, 13 November 2005 (UTC)

question: considering discrete-time processes, would the states be independent if the distribution of the values was Exponetial (or Geometric)? thanks, Akshayaj 19:56, 20 June 2006 (UTC)