- #1
schrodingerscat11
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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]