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Suppose that $X$ has a pdf $f(x)=2x$ on $[0,1]$ (and zero otherwise), and let $U$ be a uniform random variable on $[0,1]$. Find a function $g$ such that $g(U)$ and $X$ have the same distribution (behaviour).

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Is this homework? (It sounds like it.) If so, please add the homework tab. In any case, you might want to familiarize yourself with the probability integral transform. –  cardinal Jan 13 '12 at 15:08
    
sorry I'm new to the forum. Added. –  Andican Jan 13 '12 at 22:48
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Hint: Given a fixed number $t \in [0,1]$, can you figure out the value of $P\{\sqrt{U} \leq t\}$ from the information that $U$ is uniformly distributed on $[0,1]$? The answer will, of course, be a function of $t$. What is $P\{X \leq t\}$? How does it compare to $P\{\sqrt{U} \leq t\}$? After doing this (hopefully successfully), please do read about probability integral transforms as suggested by cardinal. –  Dilip Sarwate Jan 13 '12 at 23:00
    
Is there any proof about integral transformation? –  Mathematics Jan 15 '12 at 2:36
    

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