the integral PDF $\sqrt {{2 \over \pi }} \int_{t\, = \,0}^\infty {e^{\, - \,{1 \over 2}\left( {v^{\,2} /t^{\,2} + 2t} \right)} dt\,} $

Question

In my answer to this post I came to the conclusion that the PDF of the volume of a parallelepiped with normal distributed coordinates, having one vertex at the origin, is $$ p_t (v) = \sqrt {{2 \over \pi }} \int_{t\, = \,0}^\infty {e^{\, - \,{1 \over 2}\left( {v^{\,2} /t^{\,2} + 2t} \right)} dt\,} $$ with the corresponding CDF $$ P_{\,t} (v) = \int_{t = 0}^\infty {\,\,t\;e^{\, - \,\,t} \,{\rm erf}\left( {{v \over {\sqrt 2 \,t}}} \right)dt\,} $$ where $0 \le v$.

I wonder whether these two integrals may have an interesting expression in terms of known functions.

yes thanks, they could be of interest to understand the properties of this PDF — G Cab, Apr 11 '21 at 15:59
Simplification was successful there, but the proposed integrals look unchangeable... — Yuri Negometyanov, Apr 13 '21 at 13:14

Claude Leibovici · Accepted Answer · 2021-04-12T08:05:05.647

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In terms of the Meijer G function

$$p_t (v) =\frac{\sqrt{2}}{\pi }\,\, G_{0,3}^{3,0}\left(\frac{v^2}{8}| \begin{array}{c} 0,\frac{1}{2},1 \end{array} \right)$$

$$P_t (v) =\frac{2}{\pi }\,\, G_{1,4}^{3,1}\left(\frac{v^2}{8}| \begin{array}{c} 1 \\ \frac{1}{2},1,\frac{3}{2},0 \end{array} \right)$$

Edit

In terms of series, for small values of $v$ $$p_t (v)=\sqrt{\frac{2}{\pi }}-v+$$ $$\frac{v^2 \left(-2 \sqrt{2} \log (v)-2 \sqrt{2} \gamma +\sqrt{2}+\sqrt{2} \log (8)+\sqrt{2} \psi ^{(0)}\left(-\frac{1}{2}\right)\right)}{4 \sqrt{\pi }}+$$ $$\frac{v^3}{6}+O\left(v^4\right)$$ $$P_t (v)=\sqrt{\frac{2}{\pi }} v-\frac{v^2}{2}+\frac{v^3 (-6 \log (v)-9 \gamma +11+\log (8))}{18 \sqrt{2 \pi }}+O\left(v^4\right)$$

For large values of $v$, using $v=t^3$ $$p_t (v)=\frac{4t}{\sqrt{3}}\,e^{-\frac{3 t^2}{2}} \left(1+\frac{5}{18 t^2}-\frac{35}{648 t^4}+\frac{665}{34992 t^6}+O\left(\frac{1}{t^8}\right)\right)$$

$$P_t (v)=1-\frac{34}{9 \sqrt{3}}e^{-\frac{3 t^2}{2}}\left(\frac{18 t^2}{17}+1-\frac{35}{612 t^2}+\frac{1925}{33048 t^4}+O\left(\frac{1}{t^6}\right)\right)$$

edited Apr 12 '21 at 08:05

answered Apr 11 '21 at 14:37

Claude Leibovici

260,315

I see, thanks ..no hope then to get it in a simpler form. – G Cab Apr 11 '21 at 15:12
@GCab. The notion of simple or not is relative ! Back to serious, I do not think that I could do simpler. But these are just hypergeometric functions. Cheers ;-) – Claude Leibovici Apr 11 '21 at 15:15
well, of course .. what I was expecting is that they were part of a class of distributions .. I haven't heard of "Meijer's distributions" (!?) – G Cab Apr 11 '21 at 15:57
but I am starting to learn about now thanks to this paper – G Cab Apr 11 '21 at 16:06
Thanks, Claude, the asymptotic for large value is in particular interesting . – G Cab Apr 12 '21 at 13:29
@GCab. What I found interesting is how they look after $v\to t^3$ – Claude Leibovici Apr 12 '21 at 13:32
In fact, and the substitution is quite natural, since $v$ is a volume. – G Cab Apr 12 '21 at 13:34
@GCab. Amazing ! I made it to provide nicer equations. Don't repeat it : I don' t know anything in this domain (and many other, be sure). Cheers :-) – Claude Leibovici Apr 12 '21 at 13:37
@GCab My various attempts fatally leads to Meijer G function, with different parameters set. – Yuri Negometyanov Apr 13 '21 at 10:19
@YuriNegometyanov. I also think that we are in a no-end loop around various versions of Meijer G function. – Claude Leibovici Apr 13 '21 at 10:50
@YuriNegometyanov: well, it would be interesting to know the alternative parametrization. – G Cab Apr 13 '21 at 14:58
@GCab Sadlly, I was failed. Nothing new. – Yuri Negometyanov Apr 27 '21 at 10:05
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@YuriNegometyanov: thanks for your interest and effort, however. Once special functions come in, it is difficult to find simpler expressions, unless for very special values of the parameters. Meantime I came to learn that the G function is in fact involved in the product of normal variables. – G Cab Apr 27 '21 at 16:03
Is there a proof that are no simpler expressions than MeijerG? Maybe there is a Besselfunction or inserting an Error function in the integral is possible? – granular_bastard Jun 03 '21 at 10:07

the integral PDF $\sqrt {{2 \over \pi }} \int_{t\, = \,0}^\infty {e^{\, - \,{1 \over 2}\left( {v^{\,2} /t^{\,2} + 2t} \right)} dt\,} $

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