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For a physics experiment, I have a signal that is a complex number. This signal has independent rotating phases in it. phase 1: a1 and phase 2: a2 I can write my signal as a complex number

Z=F+Bexp(ia1)+cexp(ia2)

F and B are complex c is real i=sqrt(-1)

What is the expected value of Abs(Z) and of its total phase Arg(Z) knowing that a1 and a2 are uniformly distributed on the interval [0 2pi]

Is there an analytical formula for and ? How to derive it?

I supposed it has to do with functions of random variables. I've looked in math books but I feel that I won't have the time come with the answer myself.

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According to a redditor: As a1 and a2 are independent and uniformly distributed over a square, you only have to integrate the corresponding function, Z(a1, a2), times a normalization coefficient that is the area of the square.

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