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Let $X_1, X_2, \ldots X_n$ be independent Gaussian random variables with $N(0,1)$ distribution. Find the density and the characteristic function of $X_1^2 + X_2^2 + \cdots + X_n^2$

I am preparing for exams and I am stuck with this particular problem. I have seen the derivation for the characteristic function of a standard normal random variable. But I tried calculating the integral for the square and I cannot get it. Once I do that though, I think I can just raise my answer to the nth power to account for the sum. Can someone please help me with this? I have been stuck and not sure for many hours.

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Let $Y=\sum_{k=1}^n X_k^2$. The char. function of $Y$ is $\phi(t)=E(e^{itY})=\prod_{k=1}^nE(e^{itX_k^2})$, because of independence.

$E(e^{itX_k^2})=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-\frac{x^2}{2}+itx^2}dx=g(t)$. Therefore $\phi(t)=g^n(t)$.

I believe $g(t)=\frac{1}{\sqrt{1-2it}}$.

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    $\begingroup$ Thanks! How can I get the density from here? I am pretty sure it is a chi-square distribution. I am trying to do Fourier inversion $\endgroup$ – hom Dec 2 at 23:10
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    $\begingroup$ Should it be $g(t) = \frac{1}{\sqrt{1 - 2it}}$? $\endgroup$ – hom Dec 3 at 1:36
  • $\begingroup$ Yes - I got mixed up in the final step to get $g(t)$.. I corrected the answer. Getting the density function seems tough. Fourier inversion is correct, but it looks very difficult. $\endgroup$ – herb steinberg Dec 3 at 4:49
  • $\begingroup$ @hom I believe that the density function might be derivable by diirect integration. I did the first three $(Y_n=\sum_{k=1}^n X_k^2)$ and $f_n(y)$ are the density functions for$Y_n$ I got $f_1(y)=\frac{e^{-y/2}}{\sqrt{2\pi y}}$ , $f_2(y)=\frac{e^{-y/2}}{2}$, and $f_3(y)=e^{-y/2}\sqrt{\frac{y}{2\pi}}$ $\endgroup$ – herb steinberg Dec 5 at 1:39
  • $\begingroup$ @hom Since $f_2(y)$ is so simple, you can the rest of list by doing the odd and even numbered members as two lists by repeated convolutions using $f_2(y)$. $\endgroup$ – herb steinberg Dec 5 at 3:38

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