# A problem with regard to Wiener process

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.

-
@ 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
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7884342670440674, "perplexity": 664.8315631984732}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645339704.86/warc/CC-MAIN-20150827031539-00202-ip-10-171-96-226.ec2.internal.warc.gz"}