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On conditional probability of an exponential random variable

  1. Jun 26, 2012 #1
    You are given a random exponential variable X: f(x) = λ exp(-λ x).
    Suppose that X = Y + Z, where Y is the integral part of X and Z is the fractional part of X:
    Y = IP(X), Z = FP(X).
    Which is the following conditional probability:
    P(Z < z | Y = n) for 0 ≤ z < 1 and n = 0, 1, … ?
     
  2. jcsd
  3. Jun 26, 2012 #2

    haruspex

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    P(Z < z | Y = n) = P(X < n+z | Y = n) = P(X < n+z | n <= X < n+1)
    = (F(n+z)-F(n))/(F(n+1) - F(n))
    = (e-λn - e-λ(n+z))/(e-λn - e-λ(n+1))
    = (1-e-λz)/(1-e)
     
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