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Cauchy distribution is a symmetric density which equals the t distribution with one degree of freedom. The expectation and variance of the cauchy distribution do not exist. See https://en.wikipedia.org/wiki/Cauchy_distribution

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Mixture probability from mixtures of Cauchy distributions

Suppose that $Z \sim \mathrm{Bern}(p)$ and $Y_1,Y_2 \sim \mathcal{N}(0,1)$ be Gaussian random variables and let all of them be independent. Suppose I am given samples from the following random variabl …
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