The current classical theory of probability distributions have been based on the presumptive Euclidean space of parameters and random variables. In this sense the joint probability densities being derived from sequential partial derivatives of corresponding cumulative distribution function [1,2,3]. The non-Euclidean manifolds endowed with Riemannian metrics in the context of probability theory, have been introduced by many authors for phase and parameter space [4,5,6,7]. Moreover using differential geometry, some variations of information theory having been devised on non-Euclidean (Riemannian) metric spaces by Fisher, Rao, Amari and others [5,8]. In these approaches, probability distributions of various models exhibited as points on some Riemannian manifolds. By this background, differential geometry technique could be applied to analyze the probability distribution manifolds. The most applicable metric tensor on these spaces that first introduced by Fisher and Rao, is the so-called Fisher information metric [9]. Applying the Riemannian metric to define the basic concepts in statistics such as mean and covariance matrix of random variable, has also been introduced in other works [6,7,10]. In this short article we introduce a new model based on binary data arrays and matrices to analyse the variables distributions of a system of particles on the Riemannian manifold. A cloud of a large number of particles at a certain time is considered in the space of variables without analysis of its dynamical evolution and quantum physics uncertainties restrictions. The coordinates of variables in this space, is divided to infinitesimal intervals while each interval labelled by an ordered integer number. The variables of each particle occupies just one infinitesimal interval on each variable coordinate $x^{\nu }$ labelled by some integer number i. For any specified infinitesimal interval on each coordinates, there is a specific array with binary entries $\left\{0,1\right\}$ that determines the particles whose $x^{\nu }$ variable restricted to this interval. Based on this background, in sec (2) we develop the concept of particle oriented coordinates which span a flat Euclidean space on which we embed Riemannian manifolds of variable coordinates. Collection of all binary arrays of all variables, yield a binary matrix containing entire information of the particles system. By introducing a generalized inner product for a set of vectors, joint probability densities of variables, is calculated as inner product of the binary arrays that stands for some vectors in cotangent space at each point on manifold and consequently the tensor density properties of joint probability density is proved. In sec (3) based on tensor density property of the joint probabilities of variables at any point on manifold, we present a new definition for joint probability densities by symmetrized covariant derivative of cumulative distribution function. A new method to connect concepts in continuous and discrete probability theory and a novel interpretation of covariant derivative by generalized inner product has been proposed. In section (4) some examples that reveal the tensor density properties of famous physical entities are presented.

For a system consisting of a large number particles, there is a smooth and differentiable probability density function on the manifold $\mathcal{M}$ that yields the density of particles at any volume element $dV$ in manifold. This postulate is consistent with the accepted postulates of the kinetic theory of gases.

Assume a system consisting of a large number N of particles confined in an interval of space-time. Taking into account of such a system of particles, brings us the advantage of choosing a sufficient huge number of particles, moreover we could substitute particles by any kind of systems defined by their arbitrary points in parametric space. Suppose a set of independent parameters labelled by $\nu$ is to be considered in a small interval of time $\Delta \tau$. We label the particles by integer numbers up to N , and divide the possible range of each parameter into such small intervals that satisfy the accuracy of the experiment. These intervals defined as in definition 1, denoted by $\Delta x^{\nu }\left(i\right)$ where i stands for the ordered location number of an intervals on the coordinate $x^{\nu }$:

So the variable’s value of each particle falls in just one of these intervals.

For each interval $\Delta x^{\nu }\left(i\right)$, there are some particles that their variable $x^{\nu }$ places in this interval. Let record the results of these outcomes in an array whose entries are 0 or 1 in such a way that for particles with parameter value in interval $\Delta x^{\nu }\left(i\right)$,the corresponding value in array reads 1 and otherwise 0. For instance for the first particle, if its value of parameter $x^{\nu }$ falls in the range of $\Delta x^{\nu }\left(i\right)$, it returns 1, and otherwise 0. If this process iterates for all particles then we obtain an array of entries for this interval which could be arranged as a vector $e_{\nu }\left(i\right)$. This vector is a binary array which carries the information of this interval for system of particles e.g.

Each of these 0 and 1 connected to a specific particle in the system and the total number of entries equals the total number of particles N. In this example the first entry 1 means that the value of variable $x^{\nu }$ for particle with label 1 is in the ith interval. therefor these vectors are members of a vector space of dimension N. As is defined in definition 3, One may attribute to any particle, an independent basis ${\epsilon }_{\nu }$ for i th particle as where only the i th entry takes the value 1. The result of projection of a set of ${\epsilon }_{\nu }$ on the coordinate $x^{\nu }$ turns out the array $e_{\nu }\left(i\right)$ in Equation (5). This means that the sum of basis ${\epsilon }_{\nu }$ of all particles with the common value of $x^{\nu }$ yields the vector $e_{\nu }\left(i\right)$. For example the vector in Equation (5), is the sum $e_{\nu }\left(i\right)={\epsilon }_1+{\epsilon }_4+{\epsilon }_5+..$.

The vectors $e_{\nu }\left(i\right)$ at each point on $x^{\nu }$ belongs to a sub-space $\mathcal{N}$ of $\mathcal{V}$ defined in definition 3. Obviously the dimension of $\mathcal{N}$ is d. At any point P on $\mathcal{M}$, If the tangent space spanned by $\left(\frac{\partial }{\partial x^1},\frac{\partial }{\partial x^2},..,\frac{\partial }{\partial x^d}\right)$ denoted by $T_P\mathcal{M}$ and tangent space at the same point on $\mathcal{N}$ denoted by $T_P\mathcal{N}$, then there is a one to one mapping between these spaces.

Because of orthogonality, the set of $\left\{e_{\nu }\left(i\right)|1\le i\le m_{\nu }\right\}$ for a fixed $\nu$ span a tangent space $T_P^{\nu }\mathcal{N}$ which is in one to one mapping with tangent $T_P^{\nu }\mathcal{M}$ with coordinates $x^{\nu }\left(i\right)$.

Manifold $\mathcal{N}$ is an lattice Euclidean space but the manifold $\mathcal{M}$ as a lattice space, is not necessarily flat and may be endowed by a general Riemannian metric.

The probability density of particles within interval $d{x}^{\nu }\left(i\right)$ which will be denoted by $f_{\nu }\left(i\right)$, is proportional to the ratio of total number of entry "1" in the $e_{\nu }\left(i\right)$, denoted by $n_{\nu }\left(i\right)$, to total particles number N :

It is straight forward to conclude that $n_{\nu }\left(i\right)$ equals the inner product $〈e_{\nu }\left(i\right),e_{\nu }\left(i\right)〉$ . In an infinitesimal limit of $\Delta x^{\nu }\left(i\right)$ the basis $e^{*\nu }\left(i\right)$ as the dual vector of $e_{\nu }\left(i\right)$ approaches to $d{x}^{\nu }\left(i\right)$ which belongs to the cotangent (dual) space basis of $T_P^{*\nu }\mathcal{M}$ while carries the information about that interval in considered system. Respect to coordinate transformations $x^{\nu }\to {\overline{x}}^{\nu }$ in variable space we apply the rule of $d{x}^{\nu }\left(i\right)$ transformation with summation on $\nu$:

In this equation and all tensor equations in next sections, we use Einstein’s summation convention on repeated covariant and contravariant indices.

In the context of probability theory, the joint probability density of multiple (random) variables which are defined on a flat Euclidean space equipped with Cartesian coordinate, presented as a sequence of partial derivative of cumulative distribution function [1,2] and [3]:

Where F stands for cumulative distribution function (CDF) and ${\partial }_i=\frac{\partial }{\partial x^i}$ . In this sequence of partial derivatives, the repetition of indices is not allowed. Obviously $f_{ijk..}$ is not a tensor and does not meet the transformation requirement of tensors. Therefor in a flat Euclidean space with Cartesian coordinates $\left(x^1,x^2,..\right)$, joint probability density under coordinate transformation $x^{\nu }\to {\overline{x}}^{\nu }$ will transforms as:

In this section some examples of the application of joint probability densities in physics and engineering are presented. Actually the joint probability densities $f^{\mu \nu \xi ..}$ and $f_{\mu \nu \xi ..}$ as tensor densities, remind us the physical tensors such as stress-energy tensor $T^{\mu \nu }$ which stands for definitions of energy density, momentum density or energy flow density in Riemannian curved space-time manifold. Therefore these type of tensors could exemplify a physical realization of joint probability densities of two variables on the Riemannian space. the following example reveals the physical interpretation of joint probability densities as tensor density in various field of physics.

In this article a general form of joint probability density compatible with Riemannian manifold of variables has been introduced. It is shown that the joint probability densities on Riemannian manifold transforms as tensor densities of weight -1. Approach to these results facilitated by introducing a binary data matrix that collects the variables information of a system of particles on a lattice Euclidean space embedded by the particle oriented coordinates where the joint probability densities identified as a new definition of generalized (multi-linear) inner products of basis vectors. By this method the tensor density properties of joint probability is proved. Based on Taylor expansion of scalar field in a Riemannian manifolds,it has been shown that the symmetrized iterative covariant derivatives of cumulative probability function defined on Riemannian manifold, also gives the set of related joint probability densities equivalent to the generalized inner products method. As an outcome the equivalence of symmetrized iterative covariant derivatives and multi-linear inner product is proved. It has been shown the reduction of the generalized joint probability density to the usual form of iterative partial derivative of cumulative function in Euclidean space of variables with Cartesian coordinates. A new method to connect concepts in continuous and discrete probability theory and a novel interpretation of covariant derivative by generalized inner product has been proposed. Some examples of well-known physical tensors convey us that many deterministic physical variables may have fundamentally a probabilistic origin that by the future works on this subject will be more clarified.