1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability Question - Joint density function

  1. Mar 4, 2015 #1
    Before I begin, here is the question:

    If the PDF of two independent random variables X and Y are:
    f(x) = exp(-x)u(x)
    f(y) = exp(-y)u(y)
    Determine the join probability density function (JPDF) of Z&W defined by:
    Z = X+Y
    W = X/(X+Y).

    So, I know how to solve this except for one thing. How do I get the expression for Z and W.
    For Z do I literally just add f(x) + f(y) meaning z = exp(-x) + exp(-y) for (x,y) >0? Same with W?
    Once I get the expressions I just find fxy(x,y) and divide by the determinant of the Jacobian. The problem is that the Jacobian depends on the derivative of Z and W which I do not know how to get an expression for.

    Please help.
     
  2. jcsd
  3. Mar 4, 2015 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    The density function for Z is the convolution of the density functions for X and Y. For W it is much more complicated.
     
  4. Mar 4, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The formula for the density of a sum ##X+Y## of two independent random variables with densities ##f_X(x)## and ##f_Y(y)## is found in every probability textbook, as well as on-line. If you don't know it you can work it out from first principles; one way is to get density ##f_Z(z)## from the fact that
    [tex] f_Z(z) \, \Delta z \doteq P(z < X+Y < z + \Delta z) \;\text{as} \: \Delta z \to 0 [/tex]
    and to integrate the joint density ##f_X(x) f_Y(y)## over the region ##\{z < x+y < z + \Delta z \}## in ##(x,y)-## space.

    To get the joint density of ##(Z,W)##, use the standard transformation formulas that involve Jacobians, etc. The joint density of ##(X,Y)## is ##f_X(x) f_Y(y)##, and certainly does not involve a sum ##f_X(x) + f_Y(y)##, or anything like it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Probability Question - Joint density function
Loading...