Getting the joint probability density for the characteristic equation

  • #1
schrodingerscat11
89
1

Homework Statement



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.

Homework 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]

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]











 
Physics news on Phys.org
  • #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. :(
 
  • #3
Back
Top