To prove that only continuous distribution that is memoryless is exponential

Discussion in 'Scientific Statistics Math' started by beihai, Jan 9, 2004.

  1. beihai

    beihai Guest

    I am looking at a Cambrdige series textbook.

    Its proof is something like this:

    Suppose random variable T has the memoryless property.

    Let g(t)=P{T>t}

    The memoryless property can be written as g(s+t)=g(s)g(t) for all s, t
    We assume T>0 and that g(1/n)>0 for some n. Then by induction,
    g(1)=g(1/n + 1/n ... + 1/n)= [g(1/n)]^n > 0

    so g(1) = exp(-lambda) for some 0 <= lambda < infinity

    Q) May I ask how come g(1) = exp(-lambda)? Thanks for helping.

    Beihai
     
    beihai, Jan 9, 2004
    #1
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