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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.

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    $\begingroup$ 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 $\endgroup$ – wolfies Feb 1 at 14:35
  • $\begingroup$ @wolfies, Reflected Chi or Double Chi (according to the first answer)? which one is a better? $\endgroup$ – M.M Feb 1 at 15:56
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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 $\lvert X \rvert \sim\chi_k$.

For reference: Scipy calls this pdf the "double weibull", see the documentation for more details.

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