Gumbel probability distribution is an important probability distribution that has wide applications in various fields such as engineering, physics, hydrology, geology, etc. The study aims to generate a new truncated distribution by truncate the probability density function of the first type Gumbel distribution from the left side to obtain the Truncated Gumbel distribution from the left side within the period (0, ∞). As we work in this paper on the derivation of the new truncated distribution formula and its characteristics. As well as discussing different methods of estimating its parameters, represented by the maximum likelihood method and Percentiles Estimation Method, then making a simulation of the comparison between methods of estimation.
In general, the importance of cluster analysis is that one can evaluate elements by clustering multiple homogeneous data; the main objective of this analysis is to collect the elements of a single, homogeneous group into different divisions, depending on many variables. This method of analysis is used to reduce data, generate hypotheses and test them, as well as predict and match models. The research aims to evaluate the fuzzy cluster analysis, which is a special case of cluster analysis, as well as to compare the two methods—classical and fuzzy cluster analysis. The research topic has been allocated to the government and private hospitals. The sampling for this research was comprised of 288 patients being treated in 10 hospitals. As the similarity between hospitals of the study sample was measured according to the standards of quality of health services under fuzzy conditions (a case of uncertainty of the opinions of patients who were in the evaluation of health services provided to them, which was represented by a set of criteria and was measured in the form of a Likert five-point scale). Moreover, those criteria were organized into a questionnaire containing 31 items. The research found a number of conclusions, the most important is that both methods of hierarchical cluster analysis and fuzzy cluster analysis, classify the hospitals of the research sample into two clusters, each cluster comprises a group of hospitals that depend on applying health quality service standards. The second important conclusion is that the fuzzy cluster analysis is more suitable for the classification of the research sample compared to hierarchical cluster analysis.
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