# Let $f_X(x)= \frac{x^2e^{-\frac{x^2}{2}}}{\sqrt{2\pi} } 1_{(-\infty,+\infty)}$, what is this distribution called?

Let $$f_X(x)= \frac{x^2e^{-\frac{x^2}{2}}}{\sqrt{2\pi} } 1_{(-\infty,+\infty)},$$ is the pdf of random variable $$X$$. What is this distribution called?

I only know $$f_X(x)\propto x^2e^{-\frac{x^2}{2}} 1_{(0,+\infty)}$$ is the pdf of chi distribution with $$3$$ degree of freedom.

• A common nomenclature for such constructs where a known distribution with positive support is reflected around the origin is to call the new distribution a Reflected XXX. So, Reflected Chi in your case. Similarly, Reflected Weibull, Reflected Gamma etc – wolfies Feb 1 at 14:35
• @wolfies, Reflected Chi or Double Chi (according to the first answer)? which one is a better? – M.M Feb 1 at 15:56

Well, in very much the same way that you can symmetrize an exponential pdf to obtain a Laplace distribution: $$\mathrm{Lap}(x;\, b) = \begin{cases} \dfrac{1}{2}\mathrm{Exp}(x;\,\dfrac{1}{b})\quad\text{if}\;x \geq 0\\ \dfrac{1}{2}\mathrm{Exp}(-x;\, \dfrac{1}{b})\quad\text{if}\;x < 0 \end{cases}$$ you can symmetrize a chi distribution by using the same technique $$\mathrm{DoubleChi}(x;\, k) = \begin{cases} \dfrac{1}{2}\,\chi(x;\, k)\quad\text{if}\;x \geq 0\\ \dfrac{1}{2}\,\chi(-x;\, k)\quad\text{if}\;x < 0 \end{cases}$$ I'm not sure it's a very important distribution, however, you have that if $$X\sim \mathrm{DoubleChi}_k$$ then $$\lvert X \rvert \sim\chi_k$$.