_{1}

In this paper, we used two different algorithms to solve some partial differential equations, where these equations originated from the well-known two parameters of logistic distributions. The first method was the classical one that involved solving a triply of partial differential equations. The second approach was the well-known Darboux Theory. We found that the geodesic equations are a pair of isotropic curves or minimal curves. As expected, the two methods reached the same result.

In general, we confine ourselves to real geometric objects, and consequently, to real functions of real variables. Nevertheless, it is sometimes advantageous to introduce complex variables as a tool for the investigation of real surfaces. This means we should regard the real Euclidean space as being embedded in a complex Euclidean space. A curve is said to be an isotropic curve or minimal curve if the length of the arc between any two different points of the curve is zero.

Hence, a curve is isotropic if and only if

The probability density function for the logistic distribution is given by

where v is the scale parameter, and u is the location parameter.

From above equation, we derive the metric tensor components for the logistic case as follows,

Using above results, we can easily find the required tensor metric

One method to find the geodesic equation of the logistic distribution is by solving a triply of partial differential equations given in the Appendix 1 (see Struik, D.J. or Grey, A [

To avoid confusing, we only index those formulas we will use them later and ignore the other.

And the distance function is given by

It needs only two out of the three equations above to find the logistic model of geodesic equation. We will choose the Equations (1) and (3). To simplify the notation, we let

Dividing the Equation (4) by p, and integrating on both sides with respect to p, we get

Inverse Equation (5) and solve for

After substituting (6) into (3), we can derive the following results

Integrating both sides, we find the geodesic equation

where A and B are arbitrary constants

Alternatively, we can find the geodesic equation of the logistic distribution by solving one partial differential equation. This idea originated from French mathematician Darboux’s theory. A detailed proof has been given in Chen [

To solve the partial differential equation above, we may use the separable variable method as follows.

The general solution of the geodesic equation is

where A and B are arbitrary constants. This result is the same as the previous one.

The probability density function for the logistic distribution is given by

From the equation above, we derive the metric tensor components for the logistic case as follows,

The next step we need to find the moments of these partial derivatives. Some of these expectations are tricky and messy.

The first part of integral can easily check is zero since

While the second part is also zero, We can write the expectation as

The last expectation is messy and tricky.

To see the result of second part expectation,

Now, we check third part expectation,

William W. S.Chen, (2015) On Finding Geodesic Equation of Two Parameters Logistic Distribution. Applied Mathematics,06,2169-2174. doi: 10.4236/am.2015.612189

We list the six well known Christoffel Symbols as follows. For detail derivation see Struik [

In general, the solution of the geodesic equation depends upon a pair of partial differential equations as below.

For detail derivation see reference [

where

And the well known fact that