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!

Getting the joint probability density for the characteristic equation

  1. Dec 3, 2013 #1
    1. The problem statement, all variables and given/known data

    The stochastic variables X and Y are independent and Gaussian distributed with
    first moment <x> = <y> = 0 and standard deviation σx = σy = 1. Find the characteristic function
    for the random variable Z = X2+Y2, and compute the moments <z>, <z2> and <z3>. Find the first 3 cumulants.

    2. Relevant equations
    Characteristic equation: [itex]f_z (k) = <e^{ikz}> = \int_{-\infty}^{+\infty} e^{ikz}\, P_z (z) dz[/itex]

    Joint Probability density: [itex] P_z(z) = \int_{-\infty}^{+\infty} dx \, \int_{-\infty}^{+\infty} dy \, δ (z - G(x,y)) P_{x,y}(x,y) [/itex] where [itex] z = G (x, y) [/itex]

    Also, [itex]P_{x,y} = P_x (x) \, P_y (y) [/itex] for independent stochastic variables x and y.

    For Gaussian distribution: [itex] P_x = \frac{1}{\sqrt{2∏} } e^{\frac{-x^2}{2}} [/itex]

    3. The attempt at a solution
    To get the characteristic equation, we need first to get the joint probability density Pz(z):

    Since [itex] G(x,y)= x^2 +y^2 [/itex] and [itex]P_{x,y} = P_x (x) \, P_y (y) [/itex]

    [itex] P_z(z) = \int_{-\infty}^{+\infty} dx \, \int_{-\infty}^{+\infty} dy \, δ (z - x^2 +y^2) P_x (x) P_y (y) [/itex]

    [itex] P_z(z) = \int_{-\infty}^{+\infty}P_x (x) \, dx \, \int_{-\infty}^{+\infty}P_y (y) \, dy \, δ (z - x^2 +y^2) [/itex]

    [itex] P_z(z) = \int_{-\infty}^{+\infty}\frac{1}{\sqrt{2∏} } e^{\frac{-x^2}{2}} \, dx \, \int_{-\infty}^{+\infty}\frac{1}{\sqrt{2∏} } e^{\frac{-y^2}{2}} \, dy \, δ (z - x^2 +y^2) [/itex]





    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 3, 2013 #2
    I'm very sorry; I accidentally pressed the submit post button instead of preview post. How do I erase this? I'm not yet done with my post. :(
     
  4. Dec 3, 2013 #3
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted