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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](http://en.wikipedia.org/wiki/Probability_integral_transform).2012-01-13
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    sorry I'm new to the forum. Added.2012-01-13
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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.2012-01-13
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    Is there any proof about integral transformation?2012-01-15
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    http://math.stackexchange.com/questions/33922/what-exactly-is-the-probability-integral-transform2012-01-26

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