Sum of two random variables- kind of

[jmckennon]

I'm sitting here with an interesting problem that I can't seem to figure out. I'm given two random variables

X=a*exp(j*phi)
Y=b

where both a and b are known constants.

phi is uniformly distributed on the interval [0,2pi)

a third random variable Z=X+Y.

My goal, is to find the magnitude of the resulting vector.

At first, I thought that this was an easy problem that could be solved by use of convolution. This doesn't work here since phi makes X a random vector. I tried using MATLAB to help solve it. I wrote an mfile that tried solving it using convolution, and it failed. I tried turning X into a toeplitz matrix and doing matrix multiplication to do the convolution, but that too, failed.

Can anyone help me out?

 

[jmckennon]

Does anyone have an idea of how to do this?

 

[mathman]

I believe you should try to clarify what you are trying to get.

Comments: Y=b, where b is constant, so Y is not particularly random.
Z is a complex random variable, uniformly distributed over a circle of radius a, centered at b.
What do want to know any further about Z?

 

[jmckennon]

I'm essentially trying to find the pdf of Z and its' magnitude. I want to run a monte carlo simulation to verify my results and make a histogram of the results from the simulation in MATLAB for various values of A and B. The way phi is declared, (in matlab) is phi=rand(1,1000).*2*pi;

 

[jmckennon]

Does that clarify things? I appreciate the help

 

[mathman]

Z is complex so the usual concept of probability distribution [F(x)=P(X≤x)] can't be used, since X (random variable) has to be real. |Z|, being real, will have a probability distribution.

 

[jmckennon]

I understand this, but obtaining the actual solution is where I'm stuck. I'm looking for the probability density function of Z so that I can create a histogram of the values of the magnitude of Z for various a and b values.

 

[mathman]

Ordinary probability density functions are derivatives of ordinary distribution functions, which need real valued random variables. For a complex valued random variable you would need a two dimension density function treating the real and imaginary parts as (dependent) random variables.

 

[jmckennon]

I think you're mis understanding my question a bit, I'll try to clarify. In MATLAB code, phi=rand(1,1000).*2*pi; this makes X a random vector, not a random variable. If it was a random variable, things would be much easier. I'm having trouble of addressing the magnitude of the density function of a complex random vector, X that has a constant, Y being added to it. I appreciate your help though!

 

[mathman]

As far as I can tell, the density function for a random vector can only be expressed as a joint density function of its components.

 

[jmckennon]

I've made progress on this one, but I'm confused on part of the theory behind it. Here is my Matlab code.
>> a = 1;
>> b = 0.25;
>> phi = 2*pi*rand(1,10000);
>> z = a+b*exp(j*phi);
>> hist(abs(z),100)
This code produces the histogram I was looking for. It is a U shaped histogram with its' smallest value at .75, largest at 1.25 and it looks to be symmetric at 1. I'm trying to come up with an expression for the |Z| in terms of a and b.

My biggest question, and what would really help me out the most is if some one could provide like a geometric or linear algebra argument for why this
problem is relevant to the problem of the eigenvectors of two random
matrices (the area I'm tip-toeing my way into learning). I'm having trouble understanding this piece.

 

[mathman]

z=1 + 0.25(cosφ + isinφ)
|z|² = (1 + .25cosφ)² + (.25sinφ)² = 17/16 + .5cosφ

You should be able to do the rest. The min and max for |z| agree with what you observed.