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Context

Given the Probability Distribution

pdf=ProbabilityDistribution[2*(x1 - x2)*
  E^((-(x1*(3*x1 - x2)) - x2*(-x1 + 3*x2))/2)*
  Sqrt[2/Pi], {x1, -Infinity, Infinity}, 
 {x2, -Infinity, x1}];

Note the Boundary on x2.

Question

I would like to compute the Characteristic Function of this PDF.

Attempt

 CF=CharacteristicFunction[pdf, {y1, y2}]

seems to take forever before crashing the kernel.

Note: these math.SE posts link1 and link are of interest.

The purpose of knowing this Characteristic Function is to compute symbolically the matrix of scalar products involved in this question.

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With the substitutions $a=(x_1+x_2)/2$ and $b=(x_1-x_2)/2$, giving $x_1=a+b$ and $x_2=a-b$, the characteristic function $\langle e^{i(x_1t_1+x_2t_1)}\rangle$ is

Integrate[E^(I((a+b)t1+(a-b)t2)) * 2 *4*b*E^(-2(a^2+2b^2))Sqrt[2/π],
  {a, -∞, ∞}, {b, 0, ∞}]

which evaluates in a few seconds.

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  • $\begingroup$ Thanks a lot! I should have thought about this. Its puzzling why Mathematica doesn't try such transformations though. $\endgroup$ – chris Jan 9 at 13:49

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