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The beta distribution is $$ (\text{constant})\times x^{\alpha-1}(1-x)^{\beta-1} \, dx \quad\text{for } 0\le x\le 1. $$ Supposing $X$ to be so distributed, one has \begin{align} & \mu = \operatorname E X = \frac \alpha {\alpha+\beta}, \\[10pt] & \nu = \operatorname E(1-X) = \frac \beta {\alpha+\beta}, \\[4pt] & \text{(so that $\mu+\nu=1$)} \\[10pt] & \operatorname{var}(X) = \operatorname{var}(1-X) = \frac{\mu\nu} \kappa \quad \text{where } \kappa = \alpha+\beta + 1. \end{align} Thus the family of distributions is parametrized by $(\mu,\kappa)$ or by $(\nu,\kappa).$

I have on occasion called $\kappa$ a “concentration parameter.” Is there some standard name for it?

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I've heard that $\kappa := \alpha+\beta$ is sometimes also referred to as the precision of the beta distribution, precisely because if $\mu,\nu$ are fixed, then larger $\kappa$ implies lower variance. The term's meaning should carry over to your definition with the extra $+1$.

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  • $\begingroup$ It seems odd to use $\alpha+\beta$ rather than $\alpha+\beta+1,$ since if $\alpha,\beta$ both $\to0$ with the expected value fixed at $1/2$, then the variance approaches $1/4$ but $1/(\alpha+\beta) \to+\infty. \qquad$ $\endgroup$ – Michael Hardy Jun 19 '18 at 20:30

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