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Sep 2, 2014 at 16:12 comment added whuber +1. Given edits to the question, which now suggests that "bell shaped" means unimodal with a mode in the interior of the interval $[0,1]$, the first part of the analysis provides a simple direct answer: $0\lt (a-1)/(a+b-2)\lt 1$, equivalent to both $a$ and $b$ strictly greater than $1$.
Sep 2, 2014 at 5:24 history answered heropup CC BY-SA 3.0