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$X_1$ and $X_2$ are independent random variables. $X_1$ and $Y = X_1 + X_2$ have $\chi^2$ distributions with $r_1$ and $r$ degrees of freedom and $r_1 < r$.

How would I show that $X_2$ has a $\chi^2$ distribution with $r - r_1$ degrees of freedom?

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Since $X_1$ and $X_2$ are independent, the characteristic function of $Y$ is the product of the characteristic functions of $X_1$ and $X_2$, the characteristic function of a $\chi^2$ distribution with $r$ degrees of freedom is $(1-2ik)^{-r/2}$ ($k$ is theFourier variable). Thus the characteristic function of $X_2$ is $(1-2ik)^{-(r-r_1)/2}$

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