Browsing by Subject "Distribution (Probability theory)."

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    Analytic and asymptotic properties of non-symmetric Linnik's probability densities
    (1995) Erdoğan, M. Burak
    We prove that the function 1 , a 6 (0 ,2 ), ^ e R, 1 + is a characteristic function of a probability distribution if and only if ( a , 0 e P D = {{a,e) : a € (0,2), \d\ < m in (f^ , x - ^ ) (mod 27t)}. This distribution is absolutely continuous, its density is denoted by p^(x). For 0 = 0 (mod 2tt), it is symmetric and was introduced by Linnik (1953). Under another restrictions on 0 it was introduced by Laha (1960), Pillai (1990), Pakes (1992). In the work, it is proved that p^{±x) is completely monotonic on (0, oo) and is unimodal on R for any (a,0) € PD. Monotonicity properties of p^(x) with respect to 9 are studied. Expansions of p^(x) both into asymptotic series as X —»· ±oo and into conditionally convergent series in terms of log |x|, \x\^ (^ = 0 ,1 ,2 ,...) are obtained. The last series are absolutely convergent for almost all but not for all values of (a, 0) € PD. The corresponding subsets of P D are described in terms of Liouville numbers.
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    The distribution of the residual lifetime and its applications
    (1991) Çağlar, Mine Alp
    Let T be a continuous positive random variable representing the lifetime of an entitle This entity could be a human being, an animal or a plant, or a component of a mechanical or electrical system. For nonliving objects the lifetime is defined as the total amount of time for which the entitj'^ carries out its function satisfactoriljc The concept of aging involves the adverse effects of age such as increased probability of failure due to wear. In this thesis, we consider certain characteristics of the residual lifetime distribution at age t, such as the mean, median, and variance, as descril)ing aging. The following families of statistical distributions are studied from this point of view: 1. Gamma with two parameters, 2. Weil^ull with two paxameters, .3. Lognormal with two parameters, 4. Inverse Poljmomial with one parameter. Gamma and Weil)ull distrilDutions are fitted to actual data.