# Thread: Characteristic function of a product of random variables

1. ## Characteristic function of a product of random variables

Hello, does anyone know how to calculate the characteristic function of the product of two independent gaussian random variables ?
Any help welcome !
Thanks,
Jan

2. Originally Posted by NoHipHop
Hello, does anyone know how to calculate the characteristic function of the product of two independent gaussian random variables ?
Any help welcome !
Thanks,
Jan
I don't think there is a general theorem for that one (well, it is a kind of convolution, but not the usual one).

The method is the following: using the independence, we can write

$\Phi_{XY}(t)=E[e^{itXY}]=E[E[e^{itXY}|Y]]=E[\Phi_X(tY)]$

and you know $\Phi_X(t)=e^{imt-\sigma^2 t^2/2}$ (substitute with appropriate mean and variance), so you are reduced to computing $E[e^{aY+bY^2}]$ (with appropriate a and b). Write it as an integral and you probably know how to compute that (complete the square,...). The computation is easy if the mean is 0 (use the fact that the density of the Gaussian integrates to 1).