Getting the joint probability density for the characteristic equation

[schrodingerscat11]

Homework Statement

The stochastic variables X and Y are independent and Gaussian distributed with
first moment <x> = <y> = 0 and standard deviation σ_x = σ_y = 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: f_z (k) = &lt;e^{ikz}&gt; = \int_{-\infty}^{+\infty} e^{ikz}\, P_z (z) dz

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

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

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

The Attempt at a Solution

To get the characteristic equation, we need first to get the joint probability density P_z(z):

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

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

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

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)

 

[schrodingerscat11]

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. :(

 

[schrodingerscat11]

Dear viewers, my post is finally done! ^-^ Here it is.. https://www.physicsforums.com/showthread.php?t=726263

Sorry I don't know how to erase posts... :(