# A Tractable Probability Distribution with Applications in Three-Dimensional
Statistics

Seyed Mohammad Azimi-Abarghouyi The author is with the School of Electrical
Engineering and Computer Science, KTH Royal Institute of Technology,
Stockholm, Sweden (Email: seyaa@kth.se).

###### Abstract

A new family of continuous probability distributions is presented in this
paper that has applications in norm distributions of three-dimensional random
spaces. The distribution is defined on the semi-infinite interval $[0,\infty)$
and exhibits remarkable computational tractability, making it highly appealing
for various applications.

###### Index Terms:

Probability theory, continuous probability distribution, semi-infinite density
function, Euclidean norm, 3D statistics.

##  I Introduction

To date, numerous probability distributions have been introduced and utilized
in various branches of science, including the Rayleigh, Rice, and Nakagami
distributions, among others [4, 5, 6, 3, 2, 1]. For example, Rice can model
the distribution of Euclidean norms in two-dimensional random spaces.

In this work, we introduce, define, and characterize a new family of
probability distributions, including its special cases, which we name Abar
distribution111Abar means superior in Persian language, and refers to
applications of this distribution for higher dimensions than two.. Also, we
obtain its statistical measures in closed-forms. This distribution is recently
discovered in our work [7] for a research in biological communication
networks. However, to the best of our knowledge, this distribution has not
been previously discovered. For example, a similar distribution cannot be
found in websites and books on "probability and statistics" [4, 5, 6]. We
investigate use-case applications for the Abar distribution that are
applicable to three-dimensional norm or distance statistics. Our focus is on
revealing that this distribution admits a remarkably simple closed form, which
is not even the case in the simpler two-dimensional statistics. The proposed
distribution has a unique and completely different function structure that is
impossible to be extrapolated from its lower or higher dimensional cases,
which all include integrals in their functions. Using the Abar distribution,
we also find another new family of probability distributions with its special
cases, which we name Abar+ distribution. It has applications in three-
dimensional power statistics. Due to their remarkable tractability and general
applicability, the Abar and Abar+ distributions hold promise for a wide range
of applications, particularly in areas such as communication networks,
localization and signal processing, stochastic geometry, and machine learning.
By highlighting these distributions and the potential applications,
researchers will become aware of their existence and have the opportunity to
utilize them effectively.

##  II Distribution Characterization

###### Definition 1

Let $a$ and $\sigma$ two non-negative parameters. We define the Abar
distribution with the following probability density function (PDF)

$\displaystyle f_{a,\sigma}(y)=\frac{y}{\sqrt{2\pi}\sigma
a}\left[\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)-\exp\left(-\frac{(y+a)^{2}}{2\sigma^{2}}\right)\right]$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{y}{\sigma
a}\exp\left(-\frac{y^{2}+a^{2}}{2\sigma^{2}}\right)\sinh\left(\frac{ya}{\sigma^{2}}\right);\
y\geq 0,$ (1)

which is specialized for $a=0$ as

$\displaystyle f_{0,\sigma}(y)$ $\displaystyle=\lim_{a\to
0}y\frac{\frac{y-a}{\sigma^{2}}\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)+\frac{y+a}{\sigma^{2}}\exp\left(-\frac{(y+a)^{2}}{2\sigma^{2}}\right)}{\sqrt{2\pi}\sigma}$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{y^{2}}{\sigma^{3}}\exp\left(-\frac{y^{2}}{2\sigma^{2}}\right);\
y\geq 0,$ (2)

where the L’Hopital’s rule is used.222The special case $a=0$ is also found in
[7].

Further, for $a\to\infty$, (1) is specialized to

$\displaystyle f_{\infty,\sigma}(y)$
$\displaystyle=\lim_{a\to\infty}\frac{\frac{y-a}{a}+1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)$
$\displaystyle=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right);\
y\geq 0,$ (3)

which is equal to the PDF of the Gaussian distribution with mean $a$ and
variance $\sigma^{2}$, i.e., ${\cal N}(a,\sigma^{2})$.

Since
$\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)\geq\exp\left(-\frac{(y+a)^{2}}{2\sigma^{2}}\right),\forall
y$ in (1), $f_{a,\sigma}(y)\geq 0,\forall y$. In Figs 1 and 2, the defined PDF
in (1) is shown for different parameters $a$ and $\sigma$.

From (1), the cumulative distribution function (CDF) of this distribution is
obtained as

$\displaystyle F_{a,\sigma}(y)=1-\frac{1}{\sqrt{2\pi}\sigma
a}\int_{y}^{\infty}ye^{-\frac{(y-a)^{2}}{2\sigma^{2}}}-ye^{-\frac{(y+a)^{2}}{2\sigma^{2}}}\mathrm{d}y$
$\displaystyle=1-\frac{\sigma}{\sqrt{2\pi}a}\int_{y}^{\infty}\frac{y-a}{\sigma^{2}}e^{-\frac{(y-a)^{2}}{2\sigma^{2}}}-\frac{y+a}{\sigma^{2}}e^{-\frac{(y+a)^{2}}{2\sigma^{2}}}\mathrm{d}y$
$\displaystyle-\frac{1}{\sqrt{2\pi}\sigma}\int_{y}^{\infty}e^{-\frac{(y-a)^{2}}{2\sigma^{2}}}+e^{-\frac{(y+a)^{2}}{2\sigma^{2}}}\mathrm{d}y=1-\frac{\sigma}{\sqrt{2\pi}a}\times$
$\displaystyle\left(e^{-\frac{(a-y)^{2}}{2\sigma^{2}}}-e^{-\frac{(a+y)^{2}}{2\sigma^{2}}}\right)-\frac{1}{2}\biggl{(}\frac{2}{\sqrt{\pi}}\int_{-\frac{a-y}{\sqrt{2}\sigma}}^{\infty}e^{-t^{2}}\mathrm{d}t+$
$\displaystyle\frac{2}{\sqrt{\pi}}\int_{\frac{a+y}{\sqrt{2}\sigma}}^{\infty}e^{-t^{2}}\mathrm{d}t\biggr{)}=-\frac{\sigma}{\sqrt{2\pi}a}\left(e^{-\frac{(a-y)^{2}}{2\sigma^{2}}}-e^{-\frac{(a+y)^{2}}{2\sigma^{2}}}\right)$
$\displaystyle+\frac{1}{2}\left(\text{erfc}\left(\frac{a-y}{\sqrt{2}\sigma}\right)-\text{erfc}\left(\frac{a+y}{\sqrt{2}\sigma}\right)\right),$
(4)

where
$\text{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t$.
It can be observed in (II) that $\frac{a}{\sigma}$ determines the ratio of the
first part with exponential functions to the second part with erfc functions.
Also, $F_{a,\sigma}(0)=0$ and $F_{a,\sigma}(\infty)=1$, which justifies the
validity of the PDF in (1).

The mean of this distribution is

$\displaystyle\mathbb{E}_{a,\sigma}\left\\{Y\right\\}=\int_{0}^{\infty}\frac{y^{2}}{\sqrt{2\pi}\sigma
a}\Biggl{[}\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)-$
$\displaystyle\exp\left(-\frac{(y+a)^{2}}{2\sigma^{2}}\right)\Biggr{]}\mathrm{d}y=\int_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}\frac{(\sqrt{2}\sigma
t+a)^{2}}{\sqrt{\pi}a}e^{-t^{2}}\mathrm{d}t-$
$\displaystyle\int_{\frac{a}{\sqrt{2}\sigma}}^{\infty}\frac{(\sqrt{2}\sigma
t-a)^{2}}{\sqrt{\pi}a}e^{-t^{2}}\mathrm{d}t=\frac{2\sigma^{2}}{\sqrt{\pi}a}\Biggl{[}\frac{\sqrt{\pi}}{4}(1-\text{erfc}(t))-$
$\displaystyle\frac{t}{2}e^{-t^{2}}\Biggr{]}_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}+\frac{2\sqrt{2}\sigma}{\sqrt{\pi}}\left[-\frac{1}{2}e^{-t^{2}}\right]_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}+\frac{a}{\sqrt{\pi}}\times$
$\displaystyle\left[\frac{\sqrt{\pi}}{2}\left(1-\text{erfc}(t)\right)\right]_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}-\frac{2\sigma^{2}}{\sqrt{\pi}a}\Biggl{[}\frac{\sqrt{\pi}}{4}(1-\text{erfc}(t))-$
$\displaystyle\frac{t}{2}e^{-t^{2}}\Biggr{]}_{\frac{a}{\sqrt{2}\sigma}}^{\infty}+\frac{2\sqrt{2}\sigma}{\sqrt{\pi}}\left[-\frac{1}{2}e^{-t^{2}}\right]_{\frac{a}{\sqrt{2}\sigma}}^{\infty}-\frac{a}{\sqrt{\pi}}\times$
$\displaystyle\left[\frac{\sqrt{\pi}}{2}\left(1-\text{erfc}(t)\right)\right]_{\frac{a}{\sqrt{2}\sigma}}^{\infty}=\left(a+\frac{\sigma^{2}}{a}\right)\left(1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)\right)$
$\displaystyle+\sqrt{\frac{2}{\pi}}\sigma\exp\left(-\frac{a^{2}}{2\sigma^{2}}\right).$
(5)

It can be concluded from (II), when $\frac{a}{\sigma}\to 0$, we have

$\displaystyle\mathbb{E}_{a,\sigma}\left\\{Y\right\\}\to\left(a+\frac{\sigma^{2}}{a}\right)\left(1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)\right)+\sqrt{\frac{2}{\pi}}\sigma\to$
$\displaystyle\frac{1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)}{\frac{\frac{a}{\sigma^{2}}}{\frac{a^{2}}{\sigma^{2}}+1}}+\sqrt{\frac{2}{\pi}}\sigma\to\frac{1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)}{{\frac{a}{\sigma^{2}}}}+\sqrt{\frac{2}{\pi}}\sigma\stackrel{{\scriptstyle(a)}}{{\to}}$
$\displaystyle\frac{-\frac{2}{\sqrt{\pi}}\times-\frac{1}{\sqrt{2}}\exp\left(-\frac{a^{2}}{2\sigma^{2}}\right)}{\frac{1}{\sigma}}+\sqrt{\frac{2}{\pi}}\sigma\to\sqrt{\frac{2}{\pi}}\sigma+\sqrt{\frac{2}{\pi}}\sigma$
$\displaystyle=2\sqrt{\frac{2}{\pi}}\sigma,$ (6)

where $(a)$ comes from the L’Hopital’s rule. This result denotes the fact that
the terms
$\sqrt{\frac{2}{\pi}}\sigma\exp\left(-\frac{a^{2}}{2\sigma^{2}}\right)$ and
$\left(a+\frac{\sigma^{2}}{a}\right)\left(1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)\right)$
in (II) contribute the same mean when $\frac{a}{\sigma}\to 0$. Also, when
$\frac{a}{\sigma}\to\infty$, we have
$\mathbb{E}_{a,\sigma}\left\\{Y\right\\}\to
a+\frac{\sigma^{2}}{a}=a\left(1+\frac{\sigma^{2}}{a^{2}}\right)\to a$.

For the variance, first

$\displaystyle\mathbb{E}_{a,\sigma}\left\\{Y^{2}\right\\}=\int_{0}^{\infty}\frac{y^{3}}{\sqrt{2\pi}\sigma
a}\Biggl{[}\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)-$
$\displaystyle\exp\left(-\frac{(y+a)^{2}}{2\sigma^{2}}\right)\Biggr{]}\mathrm{d}y=\int_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}\frac{(\sqrt{2}\sigma
t+a)^{3}}{\sqrt{\pi}a}e^{-t^{2}}\mathrm{d}t-$
$\displaystyle\int_{\frac{a}{\sqrt{2}\sigma}}^{\infty}\frac{(\sqrt{2}\sigma
t-a)^{3}}{\sqrt{\pi}a}e^{-t^{2}}\mathrm{d}t=\frac{2\sqrt{2}\sigma^{3}}{\sqrt{\pi}a}\left[-\frac{e^{-t^{2}}}{2}\left(t^{2}+1\right)\right]_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}$
$\displaystyle+\frac{6\sigma^{2}}{\sqrt{\pi}}\left[\frac{\sqrt{\pi}}{4}(1-\text{erfc}(t))-\frac{t}{2}e^{-t^{2}}\right]_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}+\frac{3\sqrt{2}\sigma
a}{\sqrt{\pi}}\times$
$\displaystyle\left[-\frac{1}{2}e^{-t^{2}}\right]_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}+\frac{a^{2}}{\sqrt{\pi}}\left[\frac{\sqrt{\pi}}{2}\left(1-\text{erfc}(t)\right)\right]_{-\frac{a}{\sqrt{2}\sigma}}^{\infty}-\frac{2\sqrt{2}\sigma^{3}}{\sqrt{\pi}a}$
$\displaystyle\left[-\frac{e^{-t^{2}}}{2}\left(t^{2}+1\right)\right]_{\frac{a}{\sqrt{2}\sigma}}^{\infty}+\frac{6\sigma^{2}}{\sqrt{\pi}}\Biggl{[}\frac{\sqrt{\pi}}{4}(1-\text{erfc}(t))-$
$\displaystyle\frac{t}{2}e^{-t^{2}}\Biggr{]}_{\frac{a}{\sqrt{2}\sigma}}^{\infty}-\frac{3\sqrt{2}\sigma
a}{\sqrt{\pi}}\left[-\frac{1}{2}e^{-t^{2}}\right]_{\frac{a}{\sqrt{2}\sigma}}^{\infty}+$
$\displaystyle\frac{a^{2}}{\sqrt{\pi}}\left[\frac{\sqrt{\pi}}{2}\left(1-\text{erfc}(t)\right)\right]_{\frac{a}{\sqrt{2}\sigma}}^{\infty}=3\sigma^{2}+a^{2},$
(7)

which is a very simple linear function of the parameters of the PDF, $a$ and
$\sigma$, and then

$\displaystyle\mathbb{V}\text{ar}_{a,\sigma}\left\\{Y\right\\}=\mathbb{E}_{a,\sigma}\left\\{Y^{2}\right\\}-\mathbb{E}_{a,\sigma}^{2}\left\\{Y\right\\}=3\sigma^{2}+a^{2}-$
$\displaystyle\left(a+\frac{\sigma^{2}}{a}\right)^{2}\left(1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)\right)^{2}-{\frac{2}{\pi}}\sigma^{2}\exp\left(-\frac{a^{2}}{\sigma^{2}}\right)$
$\displaystyle-2\sqrt{\frac{2}{\pi}}\sigma\left(a+\frac{\sigma^{2}}{a}\right)\left(1-\text{erfc}\left(\frac{a}{\sqrt{2}\sigma}\right)\right)\exp\left(-\frac{a^{2}}{2\sigma^{2}}\right).$
(8)

When $\frac{a}{\sigma}\to 0$, we have
$\mathbb{V}\text{ar}_{a,\sigma}\left\\{Y\right\\}\to(3-{\frac{8}{\pi}})\sigma^{2}+a^{2}=\sigma^{2}\left((3-{\frac{8}{\pi}})+\frac{a^{2}}{\sigma^{2}}\right)=(3-{\frac{8}{\pi}})\sigma^{2}$.
Also, when $\frac{a}{\sigma}\to\infty$,
$\mathbb{V}\text{ar}_{a,\sigma}\left\\{Y\right\\}\to
3\sigma^{2}+a^{2}-(a+\frac{\sigma^{2}}{a})^{2}\to\sigma^{2}\left(1-\frac{\sigma^{2}}{a^{2}}\right)\to\sigma^{2}$.

Finally, the moment generating function (MGF) of this distribution is obtained
as

$\displaystyle
M_{a,\sigma}(s)=\mathbb{E}\left\\{e^{sY}\right\\}=\int_{0}^{\infty}\frac{y}{\sqrt{2\pi}\sigma
a}\Biggl{[}\exp\left(sy-\frac{(y-a)^{2}}{2\sigma^{2}}\right)$
$\displaystyle-\exp\left(sy-\frac{(y+a)^{2}}{2\sigma^{2}}\right)\Biggr{]}\mathrm{d}y=\int_{0}^{\infty}\frac{y}{\sqrt{2\pi}\sigma
a}\times$
$\displaystyle\Biggl{[}\exp\left(-\frac{1}{2\sigma^{2}}\left(y-(a+s\sigma^{2})\right)^{2}+s\left(a+\frac{s\sigma^{2}}{2}\right)\right)$
$\displaystyle-\exp\left(-\frac{1}{2\sigma^{2}}\left(y+(a-s\sigma^{2})\right)^{2}-s\left(a-\frac{s\sigma^{2}}{2}\right)\right)\Biggr{]}\mathrm{d}y=$
$\displaystyle
e^{s\left(a+\frac{s\sigma^{2}}{2}\right)}\int_{-\frac{a+s\sigma^{2}}{\sqrt{2}\sigma}}^{\infty}\frac{\sqrt{2}\sigma
t+(a+s\sigma^{2})}{\sqrt{\pi}a}e^{-t^{2}}\mathrm{d}t-e^{-s\left(a-\frac{s\sigma^{2}}{2}\right)}$
$\displaystyle\int_{\frac{a-s\sigma^{2}}{\sqrt{2}\sigma}}^{\infty}\frac{\sqrt{2}\sigma
t-(a-s\sigma^{2})}{\sqrt{\pi}a}e^{-t^{2}}\mathrm{d}t=e^{s\left(a+\frac{s\sigma^{2}}{2}\right)}\Biggl{[}-\frac{\sigma}{\sqrt{2\pi}a}\times$
$\displaystyle
e^{-t^{2}}\bigg{|}_{-\frac{a+s\sigma^{2}}{\sqrt{2}\sigma}}^{\infty}+\frac{a+s\sigma^{2}}{2a}\frac{2}{\sqrt{\pi}}\int_{-\frac{a+s\sigma^{2}}{\sqrt{2}\sigma}}^{\infty}e^{-t^{2}}\mathrm{d}t\Biggr{]}-e^{-s\left(a-\frac{s\sigma^{2}}{2}\right)}$
$\displaystyle\times\left[-\frac{\sigma}{\sqrt{2\pi}a}e^{-t^{2}}\bigg{|}_{\frac{a-s\sigma^{2}}{\sqrt{2}\sigma}}^{\infty}-\frac{a-s\sigma^{2}}{2a}\frac{2}{\sqrt{\pi}}\int_{\frac{a-s\sigma^{2}}{\sqrt{2}\sigma}}^{\infty}e^{-t^{2}}\mathrm{d}t\right]$
$\displaystyle=e^{s\left(a+\frac{s\sigma^{2}}{2}\right)}\Biggl{[}\frac{\sigma}{\sqrt{2\pi}a}e^{-\frac{(a+s\sigma^{2})^{2}}{2\sigma^{2}}}+\frac{a+s\sigma^{2}}{2a}\times$
$\displaystyle\left(2-\text{erfc}\left(\frac{a+s\sigma^{2}}{\sqrt{2}\sigma}\right)\right)\Biggr{]}-e^{-s\left(a-\frac{s\sigma^{2}}{2}\right)}\Biggl{[}\frac{\sigma}{\sqrt{2\pi}a}e^{-\frac{(a-s\sigma^{2})^{2}}{2\sigma^{2}}}$
$\displaystyle-\frac{a-s\sigma^{2}}{2a}\text{erfc}\left(\frac{a-s\sigma^{2}}{\sqrt{2}\sigma}\right)\Biggr{]}.$
(9)

It is notable that all the results in (1)-(9) are in tractable closed-forms.
This fact renders the Abar distribution extremely attractive for use in
various applications.

##  III Applications

We have discovered two applications that exhibit the Abar distribution,
specifically relating to the norm of three Gaussian random variables. In
contrast, applications involving the norm of two Gaussian random variables
result in the Rice distribution [2, 8]. Unlike the Abar distribution, the Rice
distribution cannot be expressed in a closed form and requires numerical
computations involving Bessel functions. Additionally, it possesses a distinct
structure compared to the Abar distribution. The simplicity found in the
structure of the Abar distribution is fundamentally dependent on the three-
dimensional assumption and cannot be inferred from cases with lower or higher
dimensions. In the third application, we utilize the Abar distribution to
derive yet another new tractable distribution.

Figure 1: The Abar PDF when $a=5$. Figure 2: The Abar PDF when $\sigma=5$.

###  III-A Application 1

It is presented in the following theorem.

###### Theorem 1

Assume the norm $y=\sqrt{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}$, where $y_{1}$,
$y_{2}$, and $y_{3}$ are independent Gaussian random variables with the same
variance $\sigma^{2}$ and means $a_{1}$, $a_{2}$, and $a_{3}$, respectively.
Also, let $a=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$. Then, $y$ has Abar
distribution with parameters $a$ and $\sigma$.

###### Proof:

We can write $y_{i}=z_{i}+a_{i},i\in\left\\{1,2,3\right\\}$ where
$z_{i},i\in\left\\{1,2,3\right\\}$ are independent identical Gaussian random
variables with mean zero and variance $\sigma^{2}$. Then, we have

$\displaystyle\mathbb{P}\left(\sqrt{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}<y\right)=\int_{z_{1}=-y}^{y}\int_{z_{2}=-\sqrt{y^{2}-z_{1}^{2}}}^{\sqrt{y^{2}-z_{1}^{2}}}$
$\displaystyle\int_{z_{3}=-\sqrt{y^{2}-z_{1}^{2}-z_{2}^{2}}}^{\sqrt{y^{2}-z_{1}^{2}-z_{2}^{2}}}\frac{1}{(2\pi)^{\frac{3}{2}}\sigma^{3}}\times$
$\displaystyle\exp\left(-\frac{(z_{1}-a_{1})^{2}+(z_{2}-a_{2})^{2}+(z_{3}-a_{3})^{2}}{2\sigma^{2}}\right)\mathrm{d}z_{3}\mathrm{d}z_{2}\mathrm{d}z_{1}$
$\displaystyle=\int_{z_{1}=-y}^{y}\int_{z_{2}=-\sqrt{y^{2}-z_{1}^{2}}}^{\sqrt{y^{2}-z_{1}^{2}}}\int_{z_{3}=-\sqrt{y^{2}-z_{1}^{2}-z_{2}^{2}}}^{\sqrt{y^{2}-z_{1}^{2}-z_{2}^{2}}}\frac{1}{(2\pi)^{\frac{3}{2}}\sigma^{3}}\times$
$\displaystyle\exp\left(-\frac{(z_{1}-a)^{2}+z_{2}^{2}+z_{3}^{2}}{2\sigma^{2}}\right)\mathrm{d}z_{3}\mathrm{d}z_{2}\mathrm{d}z_{1},$
(10)

where the last equality is because the Gaussian random variables are
rotationally invariant. Then, by taking derivative of (III-A), the PDF of $y$
is obtained with the help of the Leibniz integral rule and then

$\displaystyle
f_{a,\sigma}(y)=\frac{2y}{(2\pi)^{\frac{3}{2}}\sigma^{3}}\exp\left(-\frac{y^{2}+a^{2}}{2\sigma^{2}}\right)\times$
$\displaystyle\int_{z_{1}=-y}^{y}\exp\left(\frac{az_{1}}{\sigma^{2}}\right)\int_{z_{2}=-\sqrt{y^{2}-z_{1}^{2}}}^{\sqrt{y^{2}-z_{1}^{2}}}\frac{1}{\sqrt{y^{2}-z_{1}^{2}-z_{2}^{2}}}\mathrm{d}z_{2}\mathrm{d}z_{1},$
(11)

where

$\displaystyle\int_{z_{2}=-\sqrt{y^{2}-z_{1}^{2}}}^{\sqrt{y^{2}-z_{1}^{2}}}\frac{1}{\sqrt{y^{2}-z_{1}^{2}-z_{2}^{2}}}\mathrm{d}z_{2}=\pi.$
(12)

The constant equality in (12) plays the crucial role for providing a simple
closed-form PDF structure, which is not the possible case in lower or higher
dimensions (since integrals persist in the final results).

Finally, we obtain

$\displaystyle f_{a,\sigma}(y)=\frac{y}{\sqrt{2\pi}\sigma
a}\exp\left(-\frac{y^{2}+a^{2}}{2\sigma^{2}}\right)\times$
$\displaystyle\left[\exp\left(\frac{ya}{\sigma^{2}}\right)-\exp\left(-\frac{ya}{\sigma^{2}}\right)\right]=$
$\displaystyle\frac{y}{\sqrt{2\pi}\sigma
a}\left[\exp\left(-\frac{(y-a)^{2}}{2\sigma^{2}}\right)-\exp\left(-\frac{(y+a)^{2}}{2\sigma^{2}}\right)\right].$
(13)

∎

###  III-B Application 2

It is shown in our work [7, Th.1] that the distance distributions in the
three-dimensional Thomas cluster process (TCP) have the Abar
distribution.333Application 1 is the abstract form of Application 2. TCP is a
widely-used class of Poisson cluster processes in the field of stochastic
geometry and has numerous applications in various domains such as biology and
communication networks, serving as a valuable tool for modeling and analysis
purposes [8, 7, 9, 10]. As the spatial space is mostly modeled by the three
dimensions $(x,y,z)$ in these fields, the proposed PDF and results in this
work are expected to be highly valuable for spatial pattern evaluations and
performance analyses. For example, please see [7, Sections IV and V] for the
use and application in communication nanonetworks, where nanomachine sensors
are located based on three-dimensional TCP. An alternative application worth
considering is the network of clustered unmanned aerial vehicles (UAVs), where
three-dimensional TCP can be utilized for modeling the locations of UAVs in
the sky.

In addition, the distance distribution is a widely used metric in signal
processing and communication systems. For example, it can be utilized in
target localization, such as satellite localization or range detection in
radar systems, and also satellite communication systems. In these
applications, there are three location dimensions and two-dimensional modeling
is not accurate. Also, it has applications in signal (modulation)
classification and outlier detection, where distances of signals from each
other and the noise are important features.

Figure 3: The Abar+ PDF when $a=5$. Figure 4: The Abar+ PDF when $\sigma=5$.

###  III-C Application 3

The following theorem presents it.

###### Theorem 2

Assume $y={y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}$, where $y_{1}$, $y_{2}$, and
$y_{3}$ are independent Gaussian random variables with the same variance
$\sigma^{2}$ and means $a_{1}$, $a_{2}$, and $a_{3}$, respectively. Also, let
$a=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$. Then, $y$ has the following PDF with
parameters $a$ and $\sigma$

$\displaystyle
f_{a,\sigma}^{+}(y)=\frac{1}{2\sqrt{2\pi}a\sigma}\Biggl{[}\exp\left({-\frac{(a-\sqrt{y})^{2}}{2\sigma^{2}}}\right)-$
$\displaystyle\exp\left({-\frac{(a+\sqrt{y})^{2}}{2\sigma^{2}}}\right)\Biggr{]}=\frac{1}{\sqrt{2\pi}a\sigma}\exp\left(-\frac{a^{2}+y}{2\sigma^{2}}\right)\times$
$\displaystyle\sinh\left(\frac{a\sqrt{y}}{\sigma^{2}}\right);\ y\geq 0.$ (14)

###### Proof:

The CDF of $y$ is obtained as

$\displaystyle\mathbb{P}(Y<y)$
$\displaystyle=\mathbb{P}(\sqrt{Y}<\sqrt{y})=F_{a,\sigma}(\sqrt{y})$
$\displaystyle=-\frac{\sigma}{\sqrt{2\pi}a}\left(e^{-\frac{(a-\sqrt{y})^{2}}{2\sigma^{2}}}-e^{-\frac{(a+\sqrt{y})^{2}}{2\sigma^{2}}}\right)$
$\displaystyle+\frac{1}{2}\left(\text{erfc}\left(\frac{a-\sqrt{y}}{\sqrt{2}\sigma}\right)-\text{erfc}\left(\frac{a+\sqrt{y}}{\sqrt{2}\sigma}\right)\right),$
(15)

where $F_{a,\sigma}({y})$ is given in (II). Then, by taking derivative of
(III-C) and using
$\frac{\mathrm{d}}{\mathrm{d}x}\text{erfc}(x)=-\frac{2}{\sqrt{\pi}}e^{-x^{2}}$,
the proof is complete. ∎

As the PDF in (2) has not been previously discovered [4, 5, 6], we define its
distribution with the name Abar+.444This distribution is not found in [7], and
is discovered in this work. The parameters $a$ and $\sigma$ are non-negative.
In Figs 3 and 4, the defined PDF in (2) is shown for different parameters $a$
and $\sigma$. Using the L’Hopital’s rule, the special case of this
distribution for $a=0$ is given for $y\geq 0$ as

$\displaystyle f_{0,\sigma}^{+}(y)=\lim_{a\to
0}\frac{-\frac{1}{\sigma^{2}}(a-\sqrt{y})\exp\left({-\frac{(a-\sqrt{y})^{2}}{2\sigma^{2}}}\right)}{2\sqrt{2\pi}\sigma}+$
$\displaystyle\frac{\frac{1}{\sigma^{2}}(a+\sqrt{y})\exp\left({-\frac{(a+\sqrt{y})^{2}}{2\sigma^{2}}}\right)}{2\sqrt{2\pi}\sigma}=\frac{\sqrt{y}}{\sqrt{2\pi}\sigma^{3}}\exp\left(-\frac{y}{2\sigma^{2}}\right),$
(16)

which is equal to the gamma distribution with the shape $\frac{3}{2}$ and
scale ${2\sigma^{2}}$, i.e.,
$\text{Gamma}(\frac{3}{2},\frac{1}{2\sigma^{2}})$.

In addition, for $a\to\infty$, (2) goes to the all-zero function.

As signal samples can be well approximately modeled by Gaussian random
variables, where the mean represents the bias of signal, Application 3
possesses exceptional worth in the modeling and analysis of the power of
additive signals, as well as in their subsequent processing. In future
wireless multipath environments, for instance, there exist three main signal
paths between a transmitter and a receiver, which can be represented as
Gaussian random variables with different means: 1) the direct link, 2) the
reflection from ground nodes and surfaces, and 3) the reflection from sky
nodes, such as UAVs and satellites.

It is our belief that there exists a wealth of additional and potentially
unknown applications within the fields of pure and applied mathematics and
other scientific domains, specially communications and signal processing.
Furthermore, due to the remarkable tractability and closed-form structures,
the outcomes of this work hold significant potential for applications in the
areas of statistics and machine learning, particularly with regards to
classification and clustering, where norm distributions can be leveraged [12,
11]. To be more specific, clustering is a technique that groups similar data
points together based on their distance from each other. By modeling the
distances between data points, clustering algorithms can be optimized to
better identify and separate distinct clusters within the data. It is worth
mentioning that higher dimensional spaces can be visualized in three
dimensions. This is achieved through a process called dimensionality reduction
[12]. Another approach is to divide the higher dimensional space into multiple
three-dimensional spaces.

##  IV conclusions

In this work, we introduced a new probability distribution with a remarkably
simple but diverse form of PDF. Then, we derived its statistical measures, all
of which are in closed forms. Also, we provided asymptotic cases, one of which
approaches the Gaussian distribution. Furthermore, we made another new
discovery by finding another probability distribution, which includes the
gamma distribution as a special case. We highlighted diverse applications of
the proposed probability distributions in the fields of pure mathematics,
statistics and learning, stochastic geometry, signal processing, and
communication systems and networks.

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