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