1
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ 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, 2021 at 14:35
  • $\begingroup$ @wolfies, Reflected Chi or Double Chi (according to the first answer)? which one is a better? $\endgroup$
    – M.F
    Feb 1, 2021 at 15:56

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.