Is there any journal article discussing approximating a density by having a mixture density? Or even an exact conversion?
I plotted a mixture density with 2 normals ( N(0,1) and N(0,3), mean and SD in parentheses) with weights 0.5 and 0.5. It looks similar to a Cauchy(0, 1.15) although the Cauchy clearly shows longer tails.
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