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 ??