# Getting the joint probability density for the characteristic equation

1. Dec 3, 2013

### physicsjn

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: $f_z (k) = <e^{ikz}> = \int_{-\infty}^{+\infty} e^{ikz}\, P_z (z) dz$

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: $P_x = \frac{1}{\sqrt{2∏} } e^{\frac{-x^2}{2}}$

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

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

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

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

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

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. Dec 3, 2013

### physicsjn

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. Dec 3, 2013