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I know the basic idea is to generate a random number from $U(0,1)$, find the inverse cumulative distribution function $F^{-1}$ and then take $x = F^{-1}(U)$. If you were plot a histogram of say 1000 samples, the distribution would be $F$. Is there any standard technique to do point picking of a particular shape using this technique. Meaning, can you generate points uniformly inside a distribution using this technique or can you only generate a distribution of observations? The only other generalizable techniques I know of to generate points is rejection sampling.

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Yes you can uniformly pick points from an arbitrary polygon without using rejection sampling. First triangulate your polygon. Then randomly pick one of the triangles. After this, randomly choose a point inside the triangle using this. Repeat this process.

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