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

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To find the probability density function for the sum of two random variables, X and Y, where X is a complex variable uniformly distributed on a circle and Y is a constant, the Fourier transform method was initially considered. However, the discussion reveals that Z, the sum of X and Y, is uniformly distributed on a circle centered at b with radius a. The use of MATLAB to simulate Z resulted in an "upside-down Gaussian distribution," indicating a misunderstanding of the distribution's nature. The complexity arises from the fact that Z is a complex variable, not a real one, which complicates the probability density function analysis. Clarification on the nature of Z is necessary for accurate interpretation and further calculations.
jmckennon
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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?
 
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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.
 
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?
 
*upside down gaussian distribution
 
I'm confused about what you did, since Z is complex, not real.
 
I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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