How can I find the characteristic function of a bivariate gaussian distribution?

[steven187]

Hi all,

Im currently researching into Multivariate distributions, in particular I am trying to derive the characteristic function of the bivariate distribution of a gaussian. While knowing that a gaussian density function cannot be integrated how is it possible to find the characteristic function. I have been working on it but I keep bumping into a dead end. The following is what I did in the most recent dead end: (does anybody know why the latex thing aint working? I havnt been on here for a while)

so I have a double integral of the

exp{i*x*t_x+i*y*t_y}*(joint density function)

then I did the substitutions
u=x-m_x
and
v=y-m_y

I then simplified it down such that I get the following in the exponential expression

u^2*s_y^2-2*(si)*u*v*s_x*s_y+v^2*s_x^2

where si is the correlation coefficient, the problem is I can't complete the square because of the existence of si, would anybody know what I would need to do?

I had looked at Wolfram Mathworld website, its very interesting how it uses Eulers formula but isn't there any other way of doing it?

Regards

Steven

 

[mathman]

let w=au+bv and z=bu-av, where a^2+b^2=1. There will be some value of a where there will not be a cross product term for wz. Good luck!