## Calculation of probability distribution function of correlated or dependent random variables..

I am having a function ,

d^2=r^2+k1−k2*r*cos(theta) ; k1 and k2 are constants.

Where r and theta are uniformly distributed, r lies between 0 and one. and Theta lies between 0 and 2*pi.

what will be the pdf of d ?

I tried in the following way.

1) First I calculated the pdf of r^2 using transformation method. as r is uniformly distributed. the pdf of r^2 comes as 1, r^2 lies between 0 and 1. 2) Then I calculated the pdf of rcos(theta) using product formula.: for this Ifound the pdf of cos theta using transformation method and then applied product formula to find pdf of z=r^2 cos(theta). It comes as f(z)=(2*z*tan(pi/2+acos(z)))/pi, z lies between 0 and 1.

Now, how to calculate the pdf of X= r^2−k2*r*cos(theta) As, It is the difference of two random variables, I want to use sum formula (convolution theorem), but can not, because r^2 and r*cos(theta) are not independent. so how to find pdf of X for correlated random variables. ?

once the pdf of X is known, the pdf of sqrt(x+k1) can be determined using transformation method. Thus only thing is to calculate the pdf of two dependent random variables.

How to do ??

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 This is a duplicate of mathoverflow.net/questions/34738/… and I think it would be better if you updated your old question (you should be able to edit it, I think) rather than start a new version. – Yemon Choi Aug 12 2010 at 6:00