TITLE: U-shaped Beta distribution
QUESTION [0 upvotes]: I'm aware that this question is similar but for my taste, I   don't find intuitive. Why $\beta$ distribution is U shaped when $\alpha$, $\beta$ < 1?

REPLY [1 votes]: Write $\alpha^\prime:=2-\alpha,\,\beta^\prime:=2-\beta$, so a $\operatorname{B}(\alpha,\,\beta)$ distribution has PDF proportional to $x^{\alpha-1}(1-x)^{\beta-1}$, while a $\operatorname{B}(\alpha^\prime,\,\beta^\prime)$ has PDF proportional to $x^{\alpha^\prime-1}(1-x)^{\beta^\prime-1}=\frac{1}{x^{\alpha-1}(1-x)^{\beta-1}}$. In other words, the two PDFs are inversely proportional. Since $\alpha<1,\,\beta<1\implies\alpha^\prime>1,\,\beta^\prime>1$, we can take the more familiar shape for $\operatorname{B}(\alpha^\prime,\,\beta^\prime)$, then infer the shape for $\operatorname{B}(\alpha,\,\beta)$. For example, if $\alpha=\beta=\frac12$, one PDF is proportional to the $U$-shape $\frac{1}{\sqrt{x(1-x)}}$, the other to the speed bump $\sqrt{x(1-x)}$.
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