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Math Help - Discrete

  1. #1
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    Discrete

    I'm doin revision questions and I came along this question:

    CAn anyone help ?

    1. Suppose that the discrete random variable X has a geometric distribution with parameter p (0 < p < 1). In other words, suppose that
    P(X = k) = (1 - p)^(k-1) p for k = 1, 2..
    Let Y = X - 1.
    (a) Find P(Y >/ l) for each l = 0,1,2,

    (b) Show that for all nonnegative integers s and t
    P(Y >/ s + t |Y >/ s) = P(Y >/ t)
    (c) Suppose that you are working in a call centre and let Y be length of time in seconds
    that it takes you to answer a customer query.
    (i) Describe the event {Y >/ s} in words.
    (ii) Describe the result stated in (b) in words.
    (Note that the result stated in (b) is called the "memoryless property")
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  2. #2
    MHF Contributor matheagle's Avatar
    Joined
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    is >/ supposed to be \ge ?

    I haven't done this one in years...

    P(Y>a)=p\sum_{k=a+1}^{\infty}(1-p)^{k-1}

    let w=k-(a+1), so the sum starts at zero

    =p\sum_{w=0}^{\infty}(1-p)^{w+a}=p(1-p)^a\sum_{w=0}^{\infty}(1-p)^w

    =p(1-p)^a\biggl({1\over 1-(1-p)}\biggr)=(1-p)^a


    SO

    P(Y>a+b|Y>a)={P(Y>a+b)\over P(Y>a)}={(1-p)^{a+b}\over (1-p)^a}

    =(1-p)^b=P(Y>b)

    At least my memory for this memoryless property is decent.
    Last edited by matheagle; August 28th 2009 at 06:00 PM.
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