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
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- Pr(X > t + s | X > t) = Pr(X > s).
- By the definition of conditional probability, this implies
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- Pr(X > t + s)/Pr(X > t) = Pr(X > s).
- Thus we have the functional equation
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- 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)
