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If $X_1,X_2,\ldots,X_n$ be independent random variables each with distribution function $F_{X_i}$. Let $$Y:= -2\sum_{i=1}^n \ln F_{X_i}(X_i).$$

Prove that $Y$ have distribution $\chi^2(2n)$.

I'm trying solve this problem, but I can't. Any hints or tip? Thanks for your help.

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  • $\begingroup$ Meeow - my mistake. $\endgroup$ – wolfies Oct 17 '13 at 11:53
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Hints:

  1. Show that $F_X(X)$ is uniformly distributed in $(0,1)$ for any random variable $X$ with distribution function $F_X$.

  2. Show that $-2\log(U)$ follows a $\chi^2(2)$ distribution if $U$ is uniformly distributed in $(0,1)$. Note that a $\chi^2(2)$ distribution is the same as an exponential distribution with parameter $\tfrac12$.

  3. Conclude using that $X+Y\sim\chi^2(n_1+n_2)$ if $X\sim \chi^2(n_1)$ and $Y\sim\chi^2(n_2)$ are independent (this property is most easily shown using characteristic functions).

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