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Engineers commonly use the gamma distribution to describe the life span or metal fatigue of a manufactured item. In this paper, we focus on finding a geodesic equation of the two parameters gamma distribution. To find this equation, we applied both the well-known Darboux Theorem and a pair of differential equations taken from Struik [1]. The solution proposed in this note could be used as a general solution of the geodesic equation of gamma distribution. It would be interesting if we compare our results with Lauritzen’s [2].

Rao [

In general, we can use standard notation to represent the distance between two points P and Q on a curve,

However, if we can transform the distance function (2.1) to the following simplified form

it could help us to find the Geodesic Equation more easily. The task of transforming Equation (2.2) is equivalent to asking how we can determine two independent functions,

If we assume that (2.2) is valid, then it would be necessary for either the right hand side of (2.3) to be a perfect square, or for the determinant of (2.3) to be equal to zero. That is,

Equation (2.4) can be rewritten as

for convenience, we usually write the left-hand side of (2.5) as

Now, if we wish to find a general solution to (2.5), then we should rewrite (2.3) in the following form:

where both m, n are some known function of u and v.

Furthermore, if we can find an integration factor

tance function

Theorem 1: Assume the given partial differential equation

where a is an arbitrary constant. Then

Proof: See reference [

Form Section 3, we know that the coefficient of the first fundamental form is given as:

To solve the partial differential equation above, we adopt the separate variable method and

hence, form the relation

Also, form the relation

We find one of the general solutions:

Thus, by applying the Darboux Theorem, we can find the geodesic equation of the gamma distribution

where A, B are arbitrary constants.

Another method to find the geodesic equation of the gamma distribution is by solving a pair of differential equations given in the Appendix. When

and the distance function

We need only two out of three of the above equations to find our Gamma Geodesic Equation.

We will choose the first and third equations. To simplify the notation, we let

So the first equation becomes:

Dividing this equation by factor p, we get:

Thus,

Integrating on both sides, we finally get:

where A and B are arbitrary constants.

The probability density function for the gamma distribution is given by

where u and v are parameters

where

It is common to use tensor notation to E, F and G, i.e.

It is clear that E, F and G are functions of the parameters

This section lists the Christoffel Symbols of the first kind of Riemannian connection of distribution (3.1).

Following Amari [

The skewness tensors can then be calculated by using the relation from Appendix, i.e.

Whereby the

If we denote

The covariant Riemann curvature tensor, a covariant tensor of fourth order, and its Gaussian curvature can be determined as follows:

Under the assumption of distribution (3.1), we list some useful moments that may help us to derive the above tensors:

The definition of the Christoffel Symbols of the first kind in terms of the first partial derivatives of the components of the Riemannian metric tensor:

Amari [

where

with the skewness tensor

The affine connections can also be used to describe the Christoffel symbols of the second kind,

where

Next, we define the six well known Christoffel symbols (see Struik [

In case of gamma distribution and

The history of geodesic lines begins with John Bernoulli’s solution of the problem of the shortest distance between two points on a convex surface (1697-1698). In this note, our solution for the geodesic equation of gamma distribution depends on a pair of differential equations.

If we substitute the results of (3.7) into above equations, we obtain the following two equations:

By introducing the Riemann symbols of the first and second kind, respectively.

The Gaussian curvature K can be written: