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Proving the memoryless property of the exponential distribution

  1. Nov 19, 2012 #1
    Given that a random variable X follows an Exponential Distribution with paramater β, how would you prove the memoryless property?

    That is, that P(X ≤ a + b|X > a) = P(X ≤ b)

    The only step I can really think of doing is rewriting the left side as [P((X ≤ a + b) ^ (X > a))]/P(X > a). Where can I go from there?
     
  2. jcsd
  3. Nov 19, 2012 #2

    haruspex

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    P[(X ≤ a + b) ^ (X > a)] = P[X ≤ a + b] - P[X ≤ a] , right?
     
  4. Nov 20, 2012 #3
    Thanks. Using that I was able to prove it. But why is what you said true?
     
  5. Nov 20, 2012 #4

    haruspex

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    X has three disjoint ranges, <a, (a,a+b), >a+b.
    P[(X ≤ a + b) ^ (X > a)] is the probability X is in the middle range.
    P[(X ≤ a + b)] is the probability X is in the first or middle range.
    P[X ≤ a] is the probability X is in the first range.
     
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