  A new family of probability distributions $\beta_{M, N},$ $M=0\cdots N,$
$N\in\mathbb{N}$ on the unit interval $(0, 1]$ is defined by the Mellin
transform. The Mellin transform of $\beta_{M, N}$ is characterized in terms of
products of ratios of Barnes multiple gamma functions, shown to satisfy a
functional equation, and a Shintani-type infinite product factorization. The
distribution $\log\beta_{M, N}$ is infinitely divisible. If $M<N,$
$-\log\beta_{M, N}$ is compound Poisson, if $M=N,$ $\log\beta_{M, N}$ is
absolutely continuous. The integral moments of $\beta_{M, N}$ are expressed as
Selberg-type products of multiple gamma functions. The asymptotic behavior of
the Mellin transform is derived and used to prove an inequality involving
multiple gamma functions and establish positivity of a class of alternating
power series. For application, the Selberg integral is interpreted
probabilistically as a transformation of $\beta_{1, 1}$ into a product of
$\beta^{-1}_{2, 2}s.$
