# Characteristic Function of 2D Gaussian with Boole

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.

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],