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Let $B(\alpha, \beta)$ denote the Beta distribution with parameters $\alpha$ and $\beta$. What do we know about a random variable $Y \sim cB(\alpha, \beta)$ where $c \in (0, 1)$?

Obviously, $Y$ does not follow a Beta distribution anymore, as the support of the distribution of $Y$ has to be $(0, c)$ now. Does $Y$ have any well-known distribution? Is the density function of $Y$ just ''squeezed'' into $(0, c)$?

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Yes, it is just squeezed. It is a variation of non-standard beta distribution, though with non-symmetric bounds. The get the probability density function you just need to "scale back" the variable to unit range, so if $Y = cX$, where $X \sim \mathsf{Beta}(\alpha, \beta)$, you need to take $Y/c$ and pass it through the beta pdf and scale accordingly.

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