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Let $X$ be a random variable whose probability density function is

$f(x) = xe^{x-2}$, if $1 < x < 2$ and $0$ elsewhere.

Let $F(x)$ be the cumulative distribution function of $X$. Find the probability density function of the random variable $Y = F(X)$.

In this problem I integrated the distribution function to get the cumulative function of $X$ and then tried to find it's inverse but was unable to do so. Any help on how to solve problem would be appreciated.

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  • $\begingroup$ what is your integrated function? $\endgroup$ Commented Apr 14, 2016 at 13:58

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This is actually a specific instance of a more general result. In fact, you don't need to know anything about the density of $X$!

$F_Y(y) = \mathbb{P}(Y \leq y) = \mathbb{P}(F_X(X) \leq y) = \mathbb{P}(X \leq F_X^{-1}(y)) = F_X(F_X^{-1}(y)) = y$

From here you can find the density. Question: why is $F_X$ invertible?

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