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No. For a convex combination of distributions, the density is also a convex combination of the individual densities and one can easilly check that the convex combination of beta densities is not again a beta density.

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Q: Is the beta distribution closed under convex combinations viz is the convex combination of two or more beta distributions also a beta distribution?
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What is the maximum likelihood estimator of the Cauchy distribution?

Maximum likelihood estimators of the Cauchy distribution cannot be written in closed form since they are given as the roots of higher-degree polynomials. Please see the link for details.


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It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.It depends on the combination. Real numbers are closed with respect to arithmetical operations (+, -, *, /), as well as integer powers (exponents). So a combination of real numbers using any of these operators will yield a real number. But the set is not closed with respect to some fractional powers - for example, the square root of a negative number is not real.


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