LET X AND Y BE INDEPENDENT RANDOM VARIABLES AND LET Z BE A RANDOM VARIABLE (WHICH IS UNIFORM OR NOT UNIFORMLY DISTRIBUTED) OVER [X, Y]. WE STUDY THE DISTRIBUTION OF THE RANDOM VARIABLE Z AND SHOW THAT THE ARCSIN DISTRIBUTION AND CAUCHY DISTRIBUTION CAN BE CHARACTERIZED IN A PARTICULAR WAY BY MEANS OF THIS CONSTRUCTION.

IN 1975 HILL INVESTIGATED THE SPECIAL CASE OF A HEAVY TAILED ESTIMATOR FOR THE TAIL INDEX WITH PARETOTYPE TAIL. HE CONSTRUCTED A SIMPLE ESTIMATOR FOR THE TAIL INDEX BY MAXIMIZING THE CONDITIONAL LIKELIHOOD FUNCTION. IN FACT, HILL'S ESTIMATOR IS A PLOTTING PROCEDURE. IT DEPENDS ON THE NUMBERK OF LARGEST OBSERVATIONS CHOSEN TO CALCULATE THE ESTIMATE, WHEREK IS, LESS THAN THE SAMPLE SIZE N. WE CONSIDER A BAD BEHAVIOR OF THIS ESTIMATOR BY USING REAL LIFE DATA WHICH WE ENCOUNTER WHEN DATA ARE CENSORED OR IMPRECISE. IN THIS CASE, WE CALCULATE THE FIRST AND SECOND MOMENTS OF THIS POPULAR ESTIMATOR AND GIVE SOME CONDITIONS ON THE NUMBER OF ORDER STATISTICS IN ORDER TO APPLY CENTRAL LIMIT THEOREM.

THIS PAPER CONSIDERS A LINEAR PROGRAMMING PROBLEM INVOLVING RANDOM INTERVAL COEFFICIENTS. A RANDOM INTERVAL PROGRAMMING MODEL IS PRESENTED BY EXTENDING THE EXPECTATION MODEL OF STOCHASTICPROGRAMMING. THE ORIGINAL PROBLEM INVOLVING RANDOM INTERVAL PARAMETERS IS TRANSFORMED INTO A DETERMINISTIC EQUIVALENT PROBLEM USING THE PROPOSED MODEL. THE EFFICIENCY OF THE PROPOSED MODELIS CALLIED BY A NUMERICAL EXAMPLE.