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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 $U_x=\int\limits_0^1I_{\{W(t)<x\}}dt$. 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: projecteuclid.org/DPubS/Repository/1.0/… –  TheBridge Dec 13 '12 at 12:12
    
@TheBridge I'm not the owner of this question. I know that this one is a consequence of Lévy's arcsine law. –  saz Dec 13 '12 at 12:18
    
@ saz : my mistake, but I can't edit my comment. best regards. –  TheBridge Dec 13 '12 at 14:49
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