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Probability Density of Sum of Random Variables

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose X and Y are independent random variables with X following a uniform distribution on (0,1) and Y exponentially distributed with parameter [itex]\lambda = 1[/itex]. Find the density for Z = X + Y. Sketch the density and verify it integrates to 1.



    2. Relevant equations
    If Z = X + Y, and X and Y are independent,

    [tex]f_z(z) = \int_{-\infty}^{\infty}f_Y(z-x)f_X(x)\,dx[/tex]


    3. The attempt at a solution
    I am trying to catch up on stats. I think I screwed up somewhere in this problem:

    If X ~ Unif(0,1), then [itex]f_X(x)[/itex] should just be 1 on (0,1) right? Also if the parameter of the exponential distribution is 1, then [itex]f_Y(z-x) = e^{-(z-x)}?[/itex]

    But then the expression for [itex]f_z(z)[/itex] doesn't seem to make much sense. Note I have no idea what I am doing.
     
  2. jcsd
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