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## Main Question or Discussion Point

I keep reading that a random vector (X, Y) uniformly distributed over the unit circle has probability density [itex]\frac{1}{\pi}[/itex]. The only proof I've seen is that

[tex]f_{X,Y}(x,y) = \begin{cases} c, &\text{if }x^2 + y^2 \leq 1 \\ 0 &\text{otherwise}\end{cases} [/tex]

And then you solve for [itex]c[/itex] by integrating to 1. This does not seem self-evident to me. Can anyone please offer a more detailed proof? Thanks!

[tex]f_{X,Y}(x,y) = \begin{cases} c, &\text{if }x^2 + y^2 \leq 1 \\ 0 &\text{otherwise}\end{cases} [/tex]

And then you solve for [itex]c[/itex] by integrating to 1. This does not seem self-evident to me. Can anyone please offer a more detailed proof? Thanks!