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- Random variable uniform on circle. Joint distribution of coordinates?

A random variable is distributed uniformly over a circle of radius R. What does the cdf ##F(x,y)## look like as a function of the Cartesian coordinates? The pdf can be expressed as ##f(x,y)=\frac{\delta(\sqrt{x^2+y^2}-R)}{2\pi R}##, where ##\delta## is Dirac delta function. Integration is confusing.