_{1}

^{*}

The objective of this paper is to review the lifespan model. This paper will also suggest four additional general alternative computational methods not mentioned in Kass, R.E. and Vos, P.W. [1] [2]. It is not intended to compare the four formulas to be used in computing the Gaussian curvature. Four different formulas adopted from Struik, D.J. [3] are used and labeled here as (A), (B), (C), and (D). It has been found that all four of these formulas can compute the Gaussian curvature effectively and successfully. To avoid repetition, we only presented results from formulas (B) and (D). One can more easily check other results from formulas (A) and (C).

The exponential, Weibull, gamma, lognormal, inverse Gaussian, and generalized gamma distributions are the most frequently used parametric lifespan models. Among the most commonly used lifespan models, the author has chosen three that he has studied since he was a graduate student. Lawless, J.F. [

In this section, we suggest four formulas that can be used to compute the Gaussian curvature.

(C)

(D)

sum on m, where the quantities of _{11}, g_{12} and g_{22} are simply tensor notation for E, F and G. Formula (B) was developed by G. Frobenius while formula (C) was derived by J. Liouville. Clearly, formula (A) is a special case that is valid only when the parametric lines are orthogonal. Formula (D) is a general form represented in Riemann symbols of the first and second kind, respectively. In formula (D),

We can immediately calculate the same results as found from formula (A) while formula (D) results in a Riemann representation. In this way, we have supplied some more general alternative methods to compute the Gaussian curvature, including the case when

In this section, we give the needed result of derivation by applying formula (B) and (D) for computing our Gaussian curvature. The process and formulas (B) and (D) are complicated, so we decided to tabulate formulas by units, which yields some advantages. It turns out that it is much easier to check partial results than to check the whole equation. It is also much easier to understand why and how we obtain the final results, or in the event of an error it should be much easier to correct it. In model 1, we will deal with Gamma Families. In model 2, we discuss the Weibull Families. In model 3, density function is of form of Inverse Gaussian families.

Model 1: A random variable X has a gamma distribution if its probability density function is of form

Now, the information unit needed to apply the formula (B) has been available. In _{u} and F_{v} are equal zero. Using one of the fundamental properties of determinants we know that three by three’s determinant is zero. This will greatly improve the efficiency of our computation process.

In

(3.1)

Notice that the detailed results of six Christoffel symbols are given in summary

E | F | G | |||||||
---|---|---|---|---|---|---|---|---|---|

Gamma | 0 | 0 | 0 | 0 | |||||

Weibull | 0 | 0 | |||||||

Inverse Gaussian | 0 | 0 | 0 | 0 |

Δ: Determinant of formula A | ||||||
---|---|---|---|---|---|---|

Gamma | 0 | 0 | ||||

Weibull | 0 | 0 | ||||

Inverse Gaussian | 0 | 0 |

Where

Gamma | 0 | 0 | ||||

Weibull | ||||||

Inverse Gaussian | 0 | 0 |

K | |||||
---|---|---|---|---|---|

Gamma | * | ||||

Weibull | |||||

Inverse Gaussian | 0 | * |

Model 2: A random variable X has a Weibull Distribution if its probability density function is of form

We are ready to apply the formula (B). The first term involves the 3 × 3 determinant expansion. From the previous computation we aware that two terms of expansion are zero, i.e.

This means the first term of the determinant can be ignored. Also due to the fact that

Using the formula (D) to find the Weibull Distribution Gaussian curvature is our next mission. This is a somewhat messy one, as no short cut can be utilized, since two of the components of Riemann symbols have nonzero values. We show the computation as follows.

Model 3: A random variable X has an Inverse Gaussian Distribution if its probability density function is of form

Again, the information unit needed to apply the formula (B) has been available. Again, we aware that F, F_{u} and F_{v} are all equal zero. Hence the first term of determinant is zero. Also due to the fact that

(3.5)

Notice that the detailed results of six Christoffel symbols are given in summary table. Next, we apply the formula (D) to compute the curvature as below.

To summarize and compare the Gaussian curvature computed in Equations (3.1), (3.2), (3.3), (3.4), (3.5), and (3.6), it is obvious that formula (B) and (D) give us the identical results.

It is also a well-known fact that two surfaces which have the same Gaussian curvature are always isometric and bending invariant. For instance, Struik, D.J. on p. 120 provided an excellent example that demonstrated a correspondence between the points of a catenoid and that of a right helicoid, such that at corresponding points, the coefficients of the first fundamental form and the Gaussian curvatures are dentical. In fact, one surface can pass into the other by a continuous bending. This has been demonstrated by the deformation of six different stages. However, if the Gaussian curvature is different, the two surfaces will not be isometric. For example, a sphere and plane are not locally isometric because the Gaussian curvature of a sphere is nonzero while the Gaussian curvature of a plane is zero. This is why any map of a portion of the earth must distort distances. One of the most important theorems of the 19^{th} century is “Theorema Egregium”. Many mathematicians at the end of the 18^{th} century, including Euler and Monge, had used the Gaussian curvature, but only when defined as the product of the principal curvatures. Since each principal curvature of a surface depends on the particular way where the surface is defined in R^{3}, there is no obvious reason for the product of the principal curvatures to be intrinsic to that particular surface. Gauss published in 1827 that the product of the principal curvatures depends only on the intrinsic geometry of the surface revolutionized differential geometry.^{ }Gauss wrote “‘The Gaussian curvature of a surface is a bending invariant’, ‘a most excellent theorem’, ‘This is a Theorema egregium’”. In this theorem, Gauss proved that the Gaussian curvature, K, of a surface, depends only on the coefficient of the first fundamental form and their first and second derivatives. This important geometric fact will link the concepts of bending and isometric mapping.

Next, we define the six well known Christoffel symbols see Struik, D.J. or Gray A. [

we applied the following integral results

we define the nth derivative of the gamma function: