Study of laplace and related probability distributions and their applications

The aim of the present study is to investigate a probability distribution that can be derived from the laplace probability distribution and can be used to model various real world problems. In the last few decades, there has been a growing interest in the construction of flexible parametric classes of probability distributions. Various forms of the skewed and kurtotic distributions have appeared in the literature for data analysis and modeling. In particular, various forms of the skew laplace distribution have been introduced and applied in several areas including medical science, environmental science, communications, economics, engineering and finance, among others. In the present study we will investigate the skew laplace distribution based on the definition of skewed distributions introduced by O\u27Hagan and extensively studied by Azzalini. A random variable X is said to have the skew-symmetric distribution if its probability density function is f(x) = 2g(x)G(lambda x), where g and G are the probability density function and the cumulative distribution function of a symmetric distribution around 0 respectively and lambda is the skewness parameter. We will investigate the mathematical properties of this distribution and apply it to real applications. In particular, we will consider the exchange rate data for six different currencies namely, Australian Dollar,Canadian Dollar, European Euro, Japanese Yen, Switzerland Franc and United Kingdom Pound versus United States Dollar. To describe a life phenomenon we will be mostly interested when the random variableis positive. Thus, we will consider the case when the skew Laplace pdf is truncated to the left at 0 and we will study its mathematical properties. Comparisons with other life time distributions will be presented. In particular we will compare the truncated skew laplace (TSL) distribution with the two parameter Gamma probability distribution with simulated and real data with respect to its reliability behavior. We also study the hypoexponential pdf and compare it with the TSL distribution. Since the TSL pdf has increasing failure rate (IFR) we will investigate a possible application in system maintenance. In particular we study the problem related to the preventive maintenance