# A WEIBULL-NORMAL DISTRIBUTION: ITS PROPERTIES AND APPLICATIONS

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ABSTRACT

TheNormal distribution is a very important and well known probability distribution for dealing with problems in several areas of life, however there are numerous situations when the assumption of normality is not validated by the data. In this work, we propose a new model calledWeibull-Normal distribution, an extension of the normal distribution by adding two skewness parameters to the normal distribution using the Weibull generator proposed by Bourguignon et al., (2014). This study has derived explicit expressions for some of its basic statistical properties such as moments, moment generating function, the characteristics function, reliability analysis and the distribution of order statistics. The implications of the plots for the survival and hazard functions indicate that the Weibull-Normal distribution would be appropriate in modeling time or age-dependent events, where survival and failure rate decreases with time or age. Also, the plots for the pdf of the distribution showed that it is negatively skewed. The method of maximum likelihood estimation is used to estimate the parameters of our proposed model. The usefulness of the Weibull-normal distribution has been illustrated by some applications to two real data sets. The results showed that new distribution (Weibull-Normal distribution) performs better (provides better fits) than the generalizations of the normal distribution such as Kumaraswamy-Normal, Beta-Normal, Gamma-Normal, Kummer Beta-Normal and the normal distributions when the data set is negatively skewed however, the results from the second data confirmed that this distribution is more flexible and appropriate for modeling negatively skewed data sets.

CHAPTER ONE: INTRODUCTION

1.1 Background of the study

The celebrated Normal (Gaussian) distribution has been known for centuries. Its popularity has

been driven byits analytical simplicity and the associated Central Limit Theorem. The

multivariate extensionis straightforward because the marginal and conditionals are both normal,a

property rarely found in most of the other multivariate distributions. Yet there have been doubts,

reservations, and criticisms about the unqualified use of normality. There are numerous

situations when the assumption of normality is not validated by the data. In fact Geary (1947)

remarked, “Normality is a myth; there never was and never will be a normal distribution.” As an

alternative, many near normal distributions have been proposed. Some families of such near

normal distributions, which include the normal distribution and to some extent share its desirable

properties, have played a crucial role in data analysis. For description of some such families of

distributions, see Mudholkar and Hutson (2000). See also Azzalini (1985), Turner (1960) and

Prentice (1975).Many of the near norma1 distributions mentioned above deal with effects of

asymmetry. These families of asymmetrical distributions are analytically tractable, accommodate

practical values of skewness and kurtosis, and strictly include the normal distribution. These

distributions can be quite useful for data modeling and statistical analysis.

Distribution functions, their properties and interrelationships play a significant role in modeling

naturally occurring phenomena. For this reason, a large number of distribution functions which

are found applicable to many events in real life have been proposed and defined in literature.

Various methods exist in defining statistical distributions. Many of these arose from the need to

model naturally occurring events. For example, the Normal distribution addresses real-valued

1

variables that tend to cluster at a single mean value, while the Poisson distribution models

discrete rare events. Yet few other distributions are functions of one or more distributions.

Numerous standard distributions have been extensively used over the past decades for modeling

data in several fields such as Engineering, Economics, Finance, Biological, Environmental and

Medical Sciences etc. However, generalizing these standard distributions has produced several

compound distributions that are more flexible compared to the baseline distributions. For this

reason, several methods for generating new families of distributions have been studied.

1.2 Probability Distribution and Estimation Theory

In real life there is no certainty about what will happen in the future, but decisions still have to be

made. Therefore, decision processes must be able to deal with the problems of uncertainty.

Events that cannot be predicted precisely are often called random events. Many, if not most, of

the inputs to, and processes that occur in, our systems are to some extent random. Hence, so too

are the outputs or predicted impacts, and even people’s reactions to those outputs or impacts. To

ignore this randomness or uncertainty is to ignore reality. One of the commonly used tools for

dealing with uncertainty in planning and management is probability. Probability is a branch of

mathematical statistics that is used for quantitative modeling of random variables. The

probability of an event represents the proportion of times under certain conditions that the

outcome can be expected to occur. A probability density functionis a mathematical description

that approximately agrees with the frequencies or probabilities of possible events of a random

variable.

Maximum Likelihood is a popular estimation technique for many distributions because it picks

the values of the distribution's parameters that make the data more likely" than any other valueof

2

the parameter. This is accomplished by maximizing the likelihood function of the parameters

given the data. Some appealing features of Maximum Likelihood estimators include their

asymptoticunbiasedness, in that the bias tends to zero as the sample size n increases; they are

asymptotically efficient, in that they achieve the Cramer-Rao lower bound as n approaches 1;

and they are asymptotically normal.

1.3 The Weibull and the Normal Distributions

The WeibullDistribution is a very popular continuous probability distribution named after a

Swedish Engineer, Scientist and Mathematician, WaloddiWeibull (1887 – 1979). He proposed

and applied this distribution in 1939 to analyze the breaking strength of materials. Since then, it

has been widely used for analyzing li

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