In this paper, the generalized gamma (GG) distribution that is a flexible distribution in statistical literature, and has exponential, gamma, and Weibull as subfamilies, and lognormal as a limiting distribution is introduced. The power and logarithmic moments of this family is defined. A new moment estimation method of parameters of GG family using it’s characterization is presented, this method is compared with MLE method in gamma subfamily for small and large sample size. Here we study GG entropy representation and its estimation. In addition Kullback-Leibler discrimination , Akaike and Bayesian information criterion is discussed. In brief, this paper consist of presentation of general review of important properties in GG family.

We propose a dynamic program to find the shortest path in a network having gamma probability distributions as arc lengths. Two operators of sum and comparison need to be adapted for the proposed dynamic program. Convolution approach is used to sum two gamma probability distributions being employed in the dynamic program.

We consider a representation of the probability density function of a weighted convolution of the gamma distribution, where a confluent hypergeometric function describes how the differences between the parameters of the components of scale lead to departures from a density range. It is shown that the distributions can be characterized as the product between a gamma density and a confluent hypergeometric function. We give closed-form expressions for the cumulative, survival and hazard rate function. The corresponding moment generating function (m.g.f) and cumulant generating function(c.g.f) have been calculated and their properties have bean discussed.

In recent years, many researchers have attempted to estimate the soil hydraulic functions (e.g. soil moisture characteristics curve, and hydraulic conductivity function) using particle-size distribution (PSD) curve. In these studies, an accurate mathematical representation of PSD is required for fitting the observed data. So far, some mathematical models were developed with different limitations. The goodness of fit is directly related to the number of the model parameters. However, estimating the parameters for higher-parameter models which have no mathematical or physical significance is a problem. Among the current models, 2-parameter Log-normal distribution model with mathematical significant parameters has been considered as a basis for many studies. In this study, it is indicated that the 2-parameter Log-normal distribution model can not be very accurate for representation of the PSD for all of soil textural classes. As an alternative, 2-parameter Gamma distribution model is proposed for more accurate representation of the PSD that its two parameters also are mathematical significant and readily computable. These two models have been compared in fitting the observed PSD data of 461 soil samples from UNSODA soil database. Gamma distribution model indicated a pronounced improvement in representation of the PSD. Based on Coefficient of determination (R2), in 362 samples and based on RMSE, in 323 samples, Gamma distribution model showed a better representation of the PSD than Log-normal. To evaluate the significance of the difference between two models, a t-test was performed. The results showed that, at confidence level of 1%, the R2-values of the Gamma model are significantly greater than those of Log-normal model. Also, at confidence level of 5%, a significant difference between the RMSE-values of two models was shown. Therefore, 2-parameter Gamma distribution model is judged to be better than 2-parameter Log-normal model for representation of PSD.