I am trying to derive Chi-square distribution. The random variale is
$$ U^2=\sum_{i=1}^k X_i^2 $$
where $X$ is a random variable with normal standard distribution.
What is the distribution of $X^2$? I am trying to derive it using characteristic functions, but I cannot understand why $X^2$ charateristic function is:
$$\phi(t)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty e^{-\frac{x^2}{2}}e^{itx^2}dx.$$
Where did this $x^2$ in $e^{itx^2}$ come from?
Best Answer
Simply because the characteristic function of a random variable $Y$ is defined as
$$ \phi(t) = E(e^{itY}) $$
And so with $Y= X^2$ and using the density of $X \sim \mathcal{N}(0,1)$ we have $$ \phi(t) = E(e^{itY}) = E(e^{itX^2}) = \int_{-\infty}^\infty e^{itx^2} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} dx$$
Remember: When a random variable $X$ has a density $g$ then $$ E(F(X)) = \int_{-\infty}^\infty F(x) g(x) dx $$ for any $F$ measurable with $F \geq 0$ or such that $F(X)$ is integrable.