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  1. Let $X \sim U(0,1)$
    $Y=\max(X,0.5)$
    $Z=\max(X-0.5,0)$
    $W=\max(0.5-X,0)$

ask how to calculate $E(Y)$, $E(Z)$, $E(W)$

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    Hint: $$E[g(X)] = \int_{-\infty}^{\infty} g(x)f_X(x)\mathrm dx$$ where you have been told what $f_X(x)$ is. **Sketch** the function $g(x) = \max(x,0.5)$ and the function $f_X(x)$, and then the function $g(x)f_X(x)$. Compute the integral above to get $E[Y] = E[\max(X,0.5)]$. Lather, rinse, repeat for the functions $\max(X-0.5,0)$ and $\max(0.5-X,0)$.2012-03-02
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    @Dilip: +1 for *Lather, rinse, repeat*.2012-03-02
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    @Dilip Given that my probability exam just was over a few hours back, if not for your comment, I am a goner. I did not realise I had this tool at disposal!2012-03-02
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    @KannappanSampath I call this tool LOTUS (an initialism of [Law of the Unconscious Statistician](http://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician)) which helps jog my memory just when I need it most....2012-03-02
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    @DilipSarwate Now I think I'll remember this rule better. Once again, Thank you for telling me!2012-03-02

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