Finding the Probability Density Function for the Sum of Two Random Variables

[jmckennon]

Hi,

I've been working on this problem but I feel like I'm over complicating it. If you have a random variable X= a*e(j*phi), where phi is uniform on the interval [0,2pi) and a is some constant, and another random variable Y= b where b is a constant. I'm looking to find the probability density function of the random variable Z=X+Y.

This is probably really simple but from what I've been trying to do, I can just take the Fourier transform of X, Fourier transform of Y multiply them, and then take the inverse Fourier of that, but it doesn't seem to work. How can I do this?

 

[mathman]

You haven't defined j. If I can assume you mean i (sqrt(-1)), then X (complex variable) is uniformly distributed on a circle of radius a, centered at 0. Z is then uniformly distributed on a circle of radius a centered at b.

 

[jmckennon]

yes, i apologize, j is sqrt(-1). After defining in MATLAB phi=rand(1,M).*2*pi where M=1000, i plotted Z= b+a.*exp(j.*phi) for various values of a and b and it looked kinda like an upside gaussian distribution centered about pi. Is this right?

 

[jmckennon]

*upside down gaussian distribution

 

[mathman]

I'm confused about what you did, since Z is complex, not real.