There is a theorem (Khinchin/Khintchine 1938) stating that a distribution is unimodal with mode at zero iff it is the distribution function of the product of two independent random variables one of which is uniform on (0,1). I am looking for a corresponding result for a unimodal distribution with mode at c > 0. This is not a homework. Thanks a lot in advance
unimodal distribution as a product with a uniform variable
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probability-theory
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0Consider $X= U V$, where $U$ and $V$ are iid $U \sim \mathcal{U}_{(0,1)}$. Clearly $X$ is of the form you described. Yet $X>0$ with probability one, thus $X$ can not have mode at zero. Did you mean uniform on $[-1,1]$ instead? – 2012-10-16
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0[Second edit] Sasha, thank you for looking into my question. The definition of the mode allows for a discontinuity at the mode. Khinchin's (sometimes spelled as Khintchine's] theorem is for U(0,1). To clarify, I am wondering whether there exists a theorem involving a decomposition of a unimodal distribution with a mode at c>0 into a product of two independent r.v. And/or, conversely, whether a product of two independent r.v., one of which is uniform on (a,b) with a>0 produces a unimodal distribution. Many thanks :) – 2012-10-18