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Sum of independent exponential distributions with different parameters

  1. Feb 6, 2014 #1
    1. The problem statement, all variables and given/known data
    As the title indicates. I'm given two independent exponential distributions with means of 10 and 20. I need to calculate the probability that the sum of a point from each of the distributions is greater than 30.


    2. Relevant equations
    X is Exp(10)
    Y is Exp(20)

    f(x) = e^(-x/10) / 10
    g(y) = e^(-x/20) / 20


    3. The attempt at a solution

    I think I need to find the distribution of X + Y and then integrate to find the cdf. I'm not sure how to go about that though.
     
  2. jcsd
  3. Feb 7, 2014 #2

    haruspex

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    What is the probability that X lies between x and x+dx? Given that X lies in that range, what is the probability that X+Y < z (for small dx)?
     
  4. Feb 7, 2014 #3
    For small dx isn't that just the pdf of x.

    P(X+Y<z) = P(X<z-Y) = F(z-Y)
    I don't know how that helps though, assuming I'm correct.
     
  5. Feb 7, 2014 #4

    ehild

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    What is the probability that a<x a+da and b<y<b+db?

    z=x+y is a new random variable. You need the probability density f(z). The probability that z>30 comes from the cumulated distribution function F(z).

    ehild
     
  6. Feb 7, 2014 #5

    Ray Vickson

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    Yes, you need to find the pdf of X+Y. Just apply standard formulas for the density of a sum of independent random variables, in terms of the individual densities. If it is not in your textbook you can find all that you need on-line. Google 'sum of independent random variables'.
     
  7. Feb 7, 2014 #6

    haruspex

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    Almost, but dx must come into it.
    Right idea, but you can't write F(z-Y) since Y is a random variable, not a number.
    I did say "given that X lies between x and x+dx": FX+Y(z)|X=x = P(X+ Y<z | X=x) = P(Y<z-X | X=x) = FY(z-x).
    So, put those two together to get the probability that X+Y<z & x < X < x+dx.
     
  8. Feb 7, 2014 #7

    vela

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    It also might help to consider the xy-plane. What region does the condition x+y<c represent in the plane?
     
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