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.


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


 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.


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.

  • $\begingroup$ Thanks a lot! I should have thought about this. Its puzzling why Mathematica doesn't try such transformations though. $\endgroup$
    – chris
    Jan 9 '19 at 13:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.