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Question: Dealing with Elliptic Function Output

Question:Dealing with Elliptic Function Output

JoyDivisionMan 30 Maple 2023
piecewise pdf elliptic-integral randomvariable statistics
February 15 2025
0

I have constructed two Random Variables (X1 and X2).  They work the way I want.  When I take the abs(X1-X2) and ask for the PDF, I get output with Elliptic functions that seems to be in complex space. I read about the Elliptic functions, but got nowhere -- above my head.

I only want the "real" output on the 0 <= t <= 2 domain.  Does anyone know how to parse this output so that I can see a PDF as a function of t?

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NULL
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with(Statistics)
> 

fx1 := piecewise(t < -1, 0, And(t >= -1, t <= 1), 2*sqrt(-t^2+1)/Pi, t > 1, 0)
> 

fx2 := piecewise(t < -1, 0, And(t >= -1, t <= 1), 2*sqrt(-t^2+1)/Pi, t > 1, 0)
> 

X1 := RandomVariable(Distribution(PDF = unapply(fx1, t)))
> 

X2 := RandomVariable(Distribution(PDF = unapply(fx2, t)))
> 

Dist := abs(X1-X2)
> 

simplify(PDF(Dist, t))

piecewise(t <= 0, 0, t < 2, (1/3)*((-2*t^3-8*t)*EllipticE(I*t/((2+t)^(1/2)*(-t+2)^(1/2)), I*(-t^2+4)^(1/2)/t)+16*EllipticF(I*t/((2+t)^(1/2)*(-t+2)^(1/2)), I*(-t^2+4)^(1/2)/t)*t+(2*t^3+8*t)*EllipticE(I*(-t^2+4)^(1/2)/t)-16*EllipticK(I*(-t^2+4)^(1/2)/t)*t+infinity)/Pi^2, 2 <= t, 0)

 (1)
 

NULL

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