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Let $W$ be a Wiener process and $U_x$ is the amount of time spent below $x$ during time interval $(0,1)$. Hence $U_x=\int\limits_0^1I_{\{W(t). My question is: what is the probability density function of $U_x$? Thank you.

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    @ saz : This is generalisation of Paul Lévy's Arcsine law. Have a look here: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoap/10349682402012-12-13
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    @TheBridge I'm not the owner of this question. I know that this one is a consequence of Lévy's arcsine law.2012-12-13
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    @ saz : my mistake, but I can't edit my comment. best regards.2012-12-13

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