Advanced Statistics

ENCYCLOPAEDIA OF STATISTICS 1ST Edition By NICHOLAS PARAKOKWA National University of Science and Technology Zimbabwe.

CONTENTS 1. INTRODUCTION. 2. PROBABILITY THEORY. 3. DESCRETE RANDOM VARIABLES. 4. CONTINUOUS RANDOM VARIABLES. 5. VARIANCE. 6. SAMPLING. 7. POINT AND INTERVAL ESTIMATION. 8. HYPOTHESIS TESTING. 9. NONPARANETRIC TESTS. 10.LINEAR CORRELATION AND REGRESSION ANALYSIS. 11.GOODNESS-OF-FIT TESTS, CONTINGENCY TABLES. 12.TIME SERIES ANALYSIS, FORECASTING. 13.QUALITY CONTROL. 14.DECISION THEORY.

1.INTRODUCTION. STATISTICS is the mathematical science of collecting, analysing, and interpreting empirical data using inferential methods to obtain information relevant for conclusive reasoning. Although the initial study of statistics cannot be merited to a single person, great tribute is rendered to Gerolamo Cardano who, in the sixteenth century documented statistical observations. He analysed games of chance while further study continued in the seventeenth century by Pierre de Fermat and Blaise Pascal. Early statisticians concerned themselves solely with the of study discrete events which implicated combinatorial methods until analytical considerations compelled the advent of continuous variables. Andrey Nikolaevich Kolmogorov introduced the notion of sample space γ, the set of sample points xi such that xi Є γ. Richard von Misses introduced the measure theory in 1933.Contemporary texts define elementary statistical terms but we start of with probability taking it for granted that probability is in itself a more or less elementary subject whereupon terms will be defined subconsciously. Statistics has been largely divided into two sections, inferential and descriptive statistics. Inferential statistics concerns itself with the quantitative (numerical) analysis of empirical data while the latter uses qualitative analysis to provide interpretations for events. The dynamics of statistics will be illustrated to the reader who endeavours to explore an intellectual and professional cause and course in statistics.

2. PROBABILITY THEORY

2.1 There is no easy way of defining the concept probability, many a statistician prefer to describe exemplify rather than define the notion. Probability the measure of the likelihood or certainty of an event occur.

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Let xi Є γ where the xi s are sample points drawn out of the set of all possible events γ called the sample space. Then Э a δ such that 0≤δ≤1 a neighbourhood defining the certainty or likelihood of a sample point (event) xi occurring ,wherefore the measure thereof is called the probability of x denoted by P(x) subject to the following axioms: a. P(x) Є [0, 1] b. i=1n P(xi)=1 for { x1, x2,…, xn} Є γ The first axiom implies that probability ranges from 0 to 1 inclusive (can also be measured in percentage) while the latter implies that the probability of the sample space is 1. The second axiom defines the notion of collectively exhaustive sample spaces. Collectively exhaustive sample spaces. If a sample space consists of finite sample points for which each event is has constant probability of occurrence then the sum of all the probabilities of each sample point is equal to 1.Simply denoted as P(γ)=1.An obvious example of collectively exhaustive sample spaces is that of a toss of a coin. The distinct events involved in tossing a coin are obtaining a tails (T) or heads (H) for which subjectively concluded, there is a likely chance of getting either of the two hence P(T)=P(H)=.5 and P(T)+P(H)=1.

Ideally methods of ascertaining probability can be divided into two: 2.1.1 The method of subjective reasoning. The probability of an event to occur can be subject to a personal perception determined probably by an individual’s prior experience or pure guesswork. Either way the degree of an individual’s belief about the chance of an event to occur is known as the subjective probability concept. If one regards the chance of a particular racing car to finish up first as 70% in a racing tournament held once in five years due to distinctive engine efficiency another may consider the outstanding racing experience of another driver and attributes an 80% chance of a win. Subjective probabilities differ from person to person hence games of chance 2.1.2 The Relative Frequency concept. The method of relative frequency represents probability as a fraction of number of times an event occurs relative to the total number of trials. The total number of heads obtained relative to the number of trials approximates to 0, 5