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How to get the conditional distribution of $W(t/2)$ given $W(t)=x$ where $W(t)$ represents wiener process.

This was a problem in my exam and i couldn't think how to start :(

Any help!!

Thanks in advance.

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    What is stopping you in the usual approach?2012-09-02
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    $W(t)=W(t/2)+(W(t)-W(t/2))$, where the two summands are independent and normally distributed with mean $0$ and variance $t/2$. Does that help?2012-09-02
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    @HaraldHanche-Olsen Not sure this is the most direct way.2012-09-03
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    @did: I could have added that when $X$ and $Y$ are independent normally distributed variables with mean 0 and the same variance, then $X+Y$ and $X-Y$ are normal and independent as well. If you assume this known, the desired result should be right around the corner.2012-09-04

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