Association between mean residual life (MRL) and failure rate functions for continuous and discrete lifetime distributions

The purpose of this study was to correct some mistakes in the literature and derive a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. It was also desired to find the conditions under which the discrete failure rate function has an upside-down bathtub shape if corresponding MRL function has a bathtub shape. The study showed that if discrete MRL has a bathtub shape, then under some conditions the corresponding failure rate function has an upside-down bathtub shape. Also the study corrected some mistakes in proofs of Tang, Lu and Chew (1999) and established a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. Similarly, some mistakes in Gupta and Gupta (2000) are corrected, with the ensuing results being expanded and proved thoroughly to establish the relationship between the crossing points of the failure rate and associated MRL functions. The new results derived in this study will be useful to model various lifetime data that occur in environmental studies, medical research, electronics engineering, and in many other areas of science and technology.

New Bivariate Lifetime Distributions Based on Bath-Tub Shaped Failure Rate

A class of lifetime distributions which has received considerable attention in modelling and analysis of lifetime data is the class of lifetime distributions with bath-tub shaped failure rate functions because of their extensive applications. The purpose of this thesis was to introduce a new class of bivariate lifetime distributions with bath-tub shaped failure rates (BTFRFs). In this research, first we reviewed univariate lifetime distributions with bath-tub shaped failure rates, and several multivariate extensions of a univariate failure rate function. Then we introduced a new class of bivariate distributions with bath-tub shaped failure rates (hazard gradients). Specifically, the new class of bivariate lifetime distributions were developed using the method of Morgenstern’s method of defining bivariate class of distributions with given marginals. The computer simulations and numerical computations were used to investigate the properties of these distributions.

Studying Key Amino Acids in the Function of Ras Interference 1 (Rinl) SH2 and Vps9 Domains

Receptor mediated endocytosis effectively removes the "ears" with which a cell would "hear" a signal conveyed by extracellular signaling molecules, but does not necessarily block the signaling pathway in which the endocytosed receptor participates. In the process of signal attenuation, this newly formed vesicle is fused with a phagosome and the receptor molecules are degraded. Receptor mediated endocytosis as a way to attenuate epidermal growth factor (EGF) and insulin signaling will be the focus here. Ras Interference 1 (Rin 1) is a multifunctional protein involved in intracellular membrane trafficking and receptor mediated endocytosis through its Rab5 Guanine Exchange Factor and SH2 domains. The goal of this investigation is to determine the role of key amino acids involved in the interaction of Rinl with Epidermal Growth Factor Receptor and Rab5. To elucidate this role, a number of point mutations have been created and the effects of each mutation on Rin 1 function will be investigated. Key amino acids in the SH2 and Vps9 Domain were identified and effects of mutations on rate of endocytosis were observed.