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Determine probabilities involving exponential distribution

  1. Feb 13, 2015 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Problem(s):
    Suppose that X has an exponential distribution with mean equal to 10.

    Determine the following:
    (a) P(X > 10)
    (b) P(X > 20)
    (c) P(X < 30)
    (d) Find the value of x such that P(X < x) = 0.95.

    Correct answers:
    (a) 0.3679
    (b) 0.1353
    (c) 0.9502
    (d) 29.96

    2. Relevant equations
    Exponential distribution: f(x) = lambda * exp(-lambda*x) when x > 0 and 0 elsewhere (always assuming lambda > 0)

    3. The attempt at a solution
    To be honest, I'm extremely confused, and I'm stuck at part (a).

    What I'm doing is

    P(X > 10) = 1 - P(X <= x)
    P(X > 10) = 1 - integral of lambda * exp(-lambda*x) from 0 to 10 (I am integrating because I want the probability density function to be a cumulative density function)
    P(X > 10) = 1 - -[exp(-10*10) - exp(0)]
    P(X > 10) = 1 - -[exp(-100) - 1]
    P(X > 10) = 1 + [exp(-100) - exp(0)]
    P(X > 10) = 1 + exp(-100) - exp(0)
    P(X > 10) = 1 + exp(-100) - 1
    P(X > 10) = exp(-100)

    P(X > 10) = 3.72007597602083596296e-44 (which is not 0.3679)

    Any help in solving this problem would be GREATLY appreciated!
     
  2. jcsd
  3. Feb 13, 2015 #2

    LCKurtz

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    Your problem is that if the mean is ##10##, then ##\lambda = \frac 1 {10}##, not ##\lambda = 10##.
     
  4. Feb 13, 2015 #3

    s3a

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    Oh! So, my work was not complete nonsense! :D

    Mu is often the letter used to represent the mean, but what does lambda represent, though?
     
  5. Feb 13, 2015 #4

    Ray Vickson

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    It is an average rate. If we have an "arrival process" whose times of arrivals are the random epochs ##0 = T_0, T_1, T_2, T_3, \ldots##, and the successive interarrival times ##X_1 = T_1 - T_0, X_2 = T_2 - T_1, X_3 = T_3 - T_2, \ldots## are independent and exponentially distributed with parameter ##\lambda##, then the (random) number of arrivals in a time interval of length ##L## is Poisson with mean ##\lambda L##. That is, the expected number of arrivals in time ##L## is ##\lambda L##, so ##\lambda## is the expected number of arrivals per unit time = expected arrival rate.
     
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