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If you work out the convolution, the probability function will be "triangular" in shape (though still discrete of course). When you take the mod, what happens to the values $\geq q/2$? What happens to the values $<-q/2$?

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You have supplied what your inputs are, the cone and the spheric covariance, but not what your output is. Im going to guess that you want to see how far your cone is from being a unit-sphere. You make a uniform random distribution that is truly spherical and compute its covariance, then you compare that with the covariance of your cone computed through the ...

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They are both discrete uniform distributions. That is they both have constant probabilities for a finite number of values ($P(X=c) =1/k$ where $k$ is the number of values the variable can take and $c$ is a possible value of $X$.) They both have the same range and mean. They have a different $k$ (i.e., number of values the variable can take). A common ...

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The second generator is somewhat unusual, so I don't think it has its own name. Your first suggestion seems accurate but rather indirect, your second description is wrong as both are discrete, your third description does not really capture the key difference. I would just say they are both samples from a multinomial distribution. The first draws from ...

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I think if you're after a measure of uniformity, goodness of fit tests for the uniform offer a variety of statistics that can provide suitable 'uniformity' measures. If your upper and lower limits are known, Kolomogorov-Smirnov, Cramer-von Mises or Anderson-Darling statistics offer measures of uniformity (though there are a bunch of other measures available ...

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