I have 4 random variables A,B,C,D, that all have the same variance $\sigma^2$, but they are independent of each other. What I am trying to find, or to at least approximate, is the standard deviation of $\Delta$R, where $\Delta$R = $(A^2+B^2+C^2+D^2)^{1/2}$.
A,B,C,D have means $\mu_a$,$\mu_b$,$\mu_c$,$\mu_d$ respectively. The probability distribution function is:
E^(1/2 (-((x - Subscript[μ, a])^2/σ^2) - (y -
Subscript[μ, b])^2/σ^2 - (z - Subscript[μ,
c])^2/σ^2 - (u - Subscript[μ,
d])^2/σ^2))/(4 π^2 Sqrt[σ^8])
My first attempt to solve this was to convert to hyper-spherical coordinates, where
- a = $\rho$Cos($\psi$)
- b = $\rho$Sin($\psi$)Cos($\theta$)
- c = $\rho$Sin($\psi$)Sin($\theta$)Cos($\phi$)
- d = $\rho$Sin($\psi$)Sin($\theta$)Sin($\phi$)
- dV = $\rho^3$$Sin^2$($\psi$)Sin($\theta$)d$\psi$d$\theta$d$\phi$d$\rho$
and $\phi$ runs from 0 to $\pi$, $\theta$ runs from 0 to 2$\pi$, $\psi$ runs from 0 to 2$\pi$, and $\rho$ runs from 0 to r. Then $$p_r(r) = d/dr\int_0^r \int_0^{\pi} \int_0^{2\pi}\ \int_0^\pi\ PDF \rho^3Sin^2(\psi)Sin(\theta)\,d\psi\,d\theta\,d\phi\,d\rho$$
Mathematica can do the first integral and most of the second integral, but then it gets stuck.The result of the first integral is
1/3 Sin[θ] (2 B + 3 π BesselI[0, B] +
(3 π BesselI[1, A])/A -(3 π BesselI[1, B])/B +
(3 π StruveL[1, B])/B + 3 π StruveL[2, B])
Where A and B are constants, different from the ones earlier. (Sorry.) But the constants aren't super important.
A = 2$\mu_a$, and
B = 2$\mu_b$Cos($\theta$)+{2$\mu_c$$\rho$Cos($\phi$)+2$\mu_d$$\rho$Sin($\phi$)}Sin($\theta$)
The code for the first integral is below
temp1 = Sin[θ]*(Exp[A*Cos[ψ]] +
Exp[B*Sin[ψ]])*(1/(2 I)*(Exp[I*ψ] - Exp[-I*ψ]))^2
Integrate[temp1, {ψ, 0, Pi}]
Now Mathematica can do the first 4 parts of the second integral, but it cannot find the integrals of the Struve function. When I simplify the fifth part of the second integral as much as i can to
\!\(
\*SubsuperscriptBox[\(∫\), \(0\), \(2\ π\)]\(
\*FractionBox[\(π\ StruveL[1,
m\ Cos[θ] + n\ Sin[θ]]\), \(n +
m\ Cot[θ]\)] ⅆθ\)\)
Mathematica cannot compute it. Here m = 2$\mu_b$$\rho$ and n = 2$\mu_c$$\rho$Cos($\phi$)+2$\mu_d$$\rho$Sin($\phi$) . If i was able to do the second and third integral, I could then find the first and second moment to calculate the standard deviation of $\Delta$R.
So my question is, is there a way to do this quadruple integral and find $p_r(r)$? If not, is there a way to approximate or numerically calculate the value I want? Or is there a simpler/better way to find what I want?
Thanks, Kyle.
A
$Normal[\mu_a, \sigma ]$? $\endgroup$ – kglr May 23 '16 at 21:31