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If $X \sim \mathcal{N}(0,1)$, then $X^2 \sim \chi^2(1)$. What about higher powers of $X$? I know that the Gamma Distribution is a generalization of the $\chi^2$ distribution, but I don't know how the Gamma Distribution parameters relate to the square part of $\chi^2$.

In particular I'm trying to calculate $X^4$, where $X \sim \mathcal{N} \left(0,\frac{1}{N} \right)$. How do you even take on such a problem?

Thanks for any tips!

Lurco
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    @YiorgosS.Smyrlis Yet again an incorrect edit approved by you... Please revise your behaviour in this domain. – Did Feb 20 '14 at 11:54

1 Answers1

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For every positive $a$, the distribution of $T=|X|^a$ has density $$ \frac2{a\sqrt{2\pi}}t^{(1/a)-1}\exp\left(-\tfrac12t^{2/a}\right)\,\mathbf 1_{t>0}. $$ This follows from the usual change of variables method explained there.

For example, the distribution of $Z=X^4$ has density $$ \frac{1}{2\sqrt{2\pi}z^{3/4}}\exp\left(-\tfrac12\sqrt{z}\right)\,\mathbf 1_{z>0}. $$

Community
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Did
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    According to the general formula, for a=4 the exponent for z should be 1/3, not -3/4 (unless i'm missing something)? – Lurco Feb 20 '14 at 11:11
  • Read (1/a)-1, not 1/(a-1) (which would be absurd, say for a=1). – Did Feb 20 '14 at 11:52
  • Sorry for the edit, I don't know why I interpreted this exponent in such a silly way. It obviously is all correct right know. – Lurco Feb 20 '14 at 12:10
  • @Did this distribution seems not normalizable? I put it into mathematica and the integral over all $z$ is divergent. How do you explain that? – user1936752 Apr 10 '15 at 13:49
  • @user1936752 The density in my answer has integral 1, as every distribution has. – Did Apr 10 '15 at 14:02
  • Ah sorry, I realize now I need to take the absolute z if it is negative – user1936752 Apr 10 '15 at 14:21
  • @user1936752 No, the density is only defined by the function in my answer for $z>0$. For $z<0$, the density is zero, obviously. – Did Apr 10 '15 at 14:23
  • This is cool, thanks. Is there also something for X^3 (without absolute value)? – divB Aug 03 '15 at 12:22
  • @divB Indeed there is., as shows the general methodology explained there. – Did Aug 14 '15 at 16:21
  • @Did Does the distribution of $X^a$ has some name as it has for $a=2$? Maybe, it is part of some family? – Marcel May 24 '16 at 17:10
  • @Marcel One could try Generalized Gamma Distribution, see Stacy EW. A Generalization of the Gamma Distribution. The Annals of Mathematical Statistics. 1962;33(3):pp. 1187-1192. Available at: http://www.jstor.org/stable/2237889. – Did May 24 '16 at 17:14
  • @Did Do you have any references related to the sum of such variables? In particular bounds on their tails? – Thomas Ahle Dec 12 '18 at 23:16
  • @ThomasAhle If you add to it some personal input, this could make for a good new question on this site... – Did Dec 13 '18 at 07:49