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Unformatted text preview: NICHOLAS N.N. NSOWAH-NUAMAH ADVANCED TOPICS IN INTRODUCTORY PROBABILITY A FIRST COURSE IN PROBABILITY THEORY – VOLUME III Download free eBooks at bookboon.com 2 Advanced Topics In Introductory Probability: A first Course in Probability Theory – Volume III 2nd edition © 2018 Nicholas N.N. Nsowah-Nuamah & bookboon.com ISBN 978-87-403-2238-5 Download free eBooks at bookboon.com 3 ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: A FIRST COURSE IN PROBABILITY THEORY – VOLUME III Contents CONTENTS Part 1 Bivariate Probability Distributions Chapter 1 Probability And Distribution Functions 7 of Bivariate Distributions 8 1.1 Introduction 8 1.2 Concept of Bivariate Random Variables 8 1.3 Joint Probability Distributions 9 1.4 Joint Cumulative Distribution Functions 18 1.5 Marginal Distribution of Bivariate Random Variables 23 1.6 Conditional Distribution of Bivariate Random Variables 30 1.7 Independence of Bivariate Random Variables 35 Chapter 2 Sums, Differences, Products and Quotients of Bivariate Distributions 45 2.1 Introduction 45 2.2 Sums of Bivariate Random Variables 46 2.3 Differences of Random Variables 63 Fast-track your career Masters in Management Stand out from the crowd Designed for graduates with less than one year of full-time postgraduate work experience, London Business School’s Masters in Management will expand your thinking and provide you with the foundations for a successful career in business. 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For more information visit email [email protected] or call +44 (0)20 7000 7573 Download free eBooks at bookboon.com 4 Click on the ad to read more ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: A FIRST COURSE IN PROBABILITY THEORY – VOLUME III Contents 2.4 Products of Bivariate Random Variables 68 2.5 Quotients of Bivariate Random Variables 72 Chapter 3 Expectation and Variance of Bivariate Distributions 80 3.1 Introduction 80 3.2 Expectation of Bivariate Random Variables 80 3.3 Variance of Bivariate Random Variables 104 Chapter 4 Measures of Relationship of Bivariate Distributions 120 4.1 Introduction 120 4.2 Product Moment 121 4.3 Covariance of Random Variables 124 4.4 Correlation Coefficient of Random Variables 130 4.5 Conditional Expectations 135 4.6 Conditional Variances 141 4.7 Regression Curves 143 Part 2 Statistical Inequalities, Limit Laws and Sampling Distributions 154 Chapter 5 Statistical Inequalities and Limit Laws 155 5.1 Introduction 155 5.2 Markov’s Inequality 156 5.3 Chebyshev’s Inequality 160 5.4 Law of Large Numbers 170 5.5 Central Limit Theorem 177 Chapter 6 Sampling Distributions I: Basic Concepts 191 6.1 Introduction 191 6.2 Statistical Inference 192 6.3 Probability Sampling 195 6.4 Sampling With and Without Replacement 200 Chapter 7 Sampling Distributions II: Sampling Distribution of Statistics 202 7.1 Introduction 202 7.2 Sampling Distribution of Means 206 7.3 Sampling Distribution of Proportions 216 7.4 Sampling Distribution of Differences 221 7.5 Sampling Distribution of Variance 225 Download free eBooks at bookboon.com 5 ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: A FIRST COURSE IN PROBABILITY THEORY – VOLUME III Contents Chapter 8 Distributions Derived from Normal Distribution 236 8.1 Introduction 236 8.2 χ Distribution 237 8.3 t Distribution 243 8.4 F Distribution 247 Statistical Tables 254 Answers to Odd-Numbered Exercises 271 2 Bibliography 273 Download free eBooks at bookboon.com 6 ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: A FIRST COURSE IN PROBABILITY THEORY – VOLUME III Part 1 Bivariate Probability Distributions PART 1 BIVARIATE PROBABILITY DISTRIBUTIONS I salute the discovery of a single even insignificant truth more highly than all the argumentation on the highest questions which fails to reach a truth GALILEO (1564–1642) Download free eBooks at bookboon.com 7 ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: A FIRST COURSE IN PROBABILITY THEORY – VOLUME III PROBABILITY AND DISTRIBUTION FUNCTIONS OF BIVARIATE DISTRIBUTIONS Chapter 1 PROBABILITY AND DISTRIBUTION FUNCTIONS OF BIVARIATE DISTRIBUTIONS INTRODUCTION 1.1 So far, all discussions in the two volumes of my book on probability (NsowahNuamah, 2017 and 2018) have been associated with a single random variable X (that is, a one-dimensional or univariate random variable). Frequently, we may be concerned with multivariate situations that simultaneously involve two or more random variables. For instance, if we wanted to study the relationship between weight and height of individual students we might consider weight and height to be two random variables X and Y , respectively, whose values are determined by measuring the weights and heights of the students in the school. Such study will produce the ordered pair (X, Y ). 1.2 CONCEPT OF BIVARIATE RANDOM VARIABLES 1.2.1 Definition of Bivariate Random Variables Many of the concepts discussed for the one-dimensional random variables also hold for higher-dimensional case. Here in most cases we shall limit ourselves to the two-dimensional (bivariate) case; more complex multivariate 4situations are straightforward Advanced Topics in Introductory Probability generalisations. 3 Definition 1.1 BIVARIATE RANDOM VARIABLE If X = X(s) and Y = Y (s) are two real-valued functions on the sample space S, then the pair (X, Y ) that assigns a point in the real (x, y) plane to each point s ∈ S is called a bivariate random variable Synonyms of a bivariate random variable are a two-dimensional random variable/vector Fig. 1.1 is an illustration of a bivariate random variable. Fig. 1.1 1.2.2 Bivariate Random Variables Types of Bivariate Random Variables Multivariate situations, similar to univariate cases, may involve discrete, as Download free eBooks at bookboon.com well as continuous random variables. 8 ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: A FIRST COURSE IN PROBABILITY THEORY – VOLUME III Fig. 1.1 1.2.2 PROBABILITY AND DISTRIBUTION FUNCTIONS OF BIVARIATE DISTRIBUTIONS Bivariate Random Variables Types of Bivariate Random Variables Multivariate situations, similar to univariate cases, may involve discrete, as well as continuous random variables. DISCRETE BIVARIATE RANDOM VARIABLE (X, Y ) is a discrete bivariate random variable, if each of the random variables X and Y is discrete Definition 1.2 CONTINUOUS BIVARIATE RANDOM VARIABLE (X, Y ) is a continuous bivariate random variable if each of the random variables is continuous Definition 1.3 There are cases where one variable is discrete and the other continuous but this will not be considered here. Density and Distribution Functions of Bivariate Distributions 5 JOINT PROBABILITY DISTRIBUTIONS 1.3 A joint distribution is a distribution having two or more random variables, with each random variable still having its own probability distribution, expected value and variance. In addition, for ordered pair values of the random variables, probabilities will exist and the strength of any relationship between the two variables can be measured. In the multivariate case as in the univariate case we often associate a probability (mass) function with discrete random variables and a probability density function with continuous random variables. We shall take up the discrete case first since it is the easier one to understand. 1.3.1 Joint Probability Distribution of Discrete Random Variables Suppose that X and Y are discrete random variables, and X takes values i = 0, 1, 2, · · · , n, and Y takes values j = 1, 2, · · · , m. Most often, such a joint distribution is given in table form. Table 1.1 is an n-by-m array which displays the number of occurrences of the various combinations of values of X and Y . We may observe that each row represents values of X and each column represents values of Y . The row and column totals are called marginal totals. Such a table is called the joint frequency distribution. Table 1.1 Joint Frequency Distribution of X and Y X x1 x2 .. . Row Totals Y y1 y2 (x1 , y1 ) (x1 , y2 ) (x2 , y1 ) .. . (x2 , y2 ) .. . ··· ym ··· (x1 , ym ) ··· (x2 , ym )  x1  x2 y y Download free eBooks at bookboon.com .. . .. . 9 .. . i = 0, 1, 2, · · · , n, and Y takes values j = 1, 2, · · · , m. Most often, such a joint distribution is given in table form. Table 1.1 is an n-by-m array which displays the number of occurrences of the various combinations of values ADVANCED TOPICS IN INTRODUCTORY PROBABILITY: FIRST COURSE IN that each row represents PROBABILITY FUNCTIONS of X and Y . A We may observe values ofAND X DISTRIBUTION and PROBABILITY – VOLUME BIVARIATE DISTRIBUTIONS each column THEORY represents values IIIof Y . The row and column totals areOFcalled marginal totals. Such a table is called the joint frequency distribution. Table 1.1 Joint Frequency Distribution of X and Y X x1 x2 y2 (x1 , y1 ) (x1 , y2 ) (x2 , y1 ) .. . xn Column Totals 6 Row Totals Y y1 (x2 , y2 ) .. . .. . (xn , y1 ) (xn , y2 )  y1 x  ··· ym ··· (x1 , ym ) ··· (x2 , ym ) .. . .. . ··· (xn , ym ) x x1  x2  xn y  ··· y2  .. . y y ym x  x xi yj = N y Advanced Topics in Introductory Probability For example, suppose X and Y are discrete random variables, and X takes values 0, 1, 2, 3, and Y takes values 1, 2, 3. Each of the nm row-column intersections in Table 1.2 represents the frequency that belongs to the ordered pair (X, Y ). Table 1.2 Joint Frequency Distribution of X and Y Values of X 0 1 2 3 Column Totals Values of Y 1 1 0 0 1 2 2 0 2 2 0 4 3 0 1 1 0 2 Row Totals 1 3 3 1 8 Definition 1.4 JOINT PROBABILITY DISTRIBUTION Let X and Y be discrete random variables with possible values xi , i = 1, 2, ..., n and yj , j = 1, 2, 3, ..., m, respectively. The joint (or bivariate) probability distribution for X and Y is given by p(xi , yj ) = P ({X = xi } ∩ {Y = yj }) defined for all (xi , yj ) The function p(xi , yj ) is sometimes referred to as the joint probability mass function (p.m.f.) or the joint probability function (p.f.) of X and Y . This function gives the probability that X will assume a particular value x while at the same time Y will assume a particular value y. Note (a) The notation p(x, y) for all (x, y) is the same as writing p(xi , yj ) for eBooks at bookboon.com i = 1, 2, ..., n and j = 1, 2,Download 3, ..., m.free Sometimes when there is no ambi10 guity we shall use simply p(x, y). The function p(xi , yj ) is sometimes referred to as the joint probability ADVANCED TOPICS IN INTRODUCTORY mass functionA (p.m.f.) or theINjoint probability function (p.f.) of XAND andDISTRIBUTION Y. PROBABILITY: FIRST COURSE PROBABILITY FUNCTIONS PROBABILITY – VOLUME III BIVARIATE DISTRIBUTIONS This functionTHEORY gives the probability that X will assume a particular OF value x while at the same time Y will assume a particular value y. Note (a) The notation p(x, y) for all (x, y) is the same as writing p(xi , yj ) for i = 1, 2, ..., n and j = 1, 2, 3, ..., m. Sometimes when there is no ambiDensity andwe Distribution Functions of Bivariate Distributions 7 guity shall use simply p(x, y). (b) The joint probability p(xi , yj ) is sometimes denoted as P (X = x, Y = y), where the comma stands for ‘and’ or ‘∩’. Definition 1.5 If X and Y are discrete random variables with joint probability mass function p(xi , yj ), then (a) p(xi , yj ) ≥ 0, (b) n  m  for all i and j p(xi , yj ) = 1 i=1 j=1 Once the joint probability mass function is determined for discrete random variables X and Y , calculation of joint probabilities involving X and Y is straightforward. Let the value that the random variables X and Y jointly take be denoted by the ordered pair (xi , yj ). The joint probability p(xi , yj ) is obtained by counting the number of occurrences of that combination of values X and Y and dividing the count by the total number of all the sample points. Thus, P ({X = xi } ∩ {Y = yj }) = #({X = xi } ∩ {Y = yj }) n  m  i=1 j=1 = #({X = xi } ∩ {Y = yj }) #(xi , yj ) n  m  #(xi , yj ) i=1 j=1 where #(xi , yj ) is the number of occurrences in the cell of the ordered pair (xi , yj ); n  m  #(xi , yj ) is the total number of all sample points (cells) i=1 j=1 of the ordered pairs (xi , yj ), denoted by N . Download free eBooks at bookboon.com 11 ADVANCED TOPICS IN INTRODUCTORY 8 PROBABILITY: A FIRST COURSE INAdvanced PROBABILITY THEORY – VOLUME III Topics in Introductory Probability PROBABILITY AND DISTRIBUTION FUNCTIONS OF BIVARIATE DISTRIBUTIONS Joint Probability Distribution of Bivariate Random Variables in Tabular Form The joint probability distribution may be given in the form of a table of n rows and m columns (See Table 1.3). The upper margins of the table indicate the possible distinct values of X and Y . The numbers in the body of the table are the probabilities for the joint occurrences of the two events corresponding to X = xi (1 ≤ i ≤ n) and Y = yj (1 ≤ i ≤ m). The row and column totals are the probabilities for the individual random variables and are called marginal probabilities because they appear on the margins of the table. Such a table is also called the joint relative frequency distribution. Table 1.3 Joint Probability Distribution of X and Y X x1 x2 .. . y1 p(x1 , y1 ) p(x2 , y1 ) .. . Y y2 p(x1 , y2 ) p(x2 , y2 ) .. . xn p(xn , y1 ) p(y1 ) Column Totals Row Totals ··· ··· ··· .. . ym p(x1 , ym ) p(x2 , ym ) .. . p(x1 ) p(x2 ) .. . p(xn , y2 ) ··· p(xn , ym ) p(xn ) p(y2 ) ··· p(ym ) n  m  p(xi , yj ) = 1 i=1 j=1 Note The marginal probabilities for X are simply the simple probabilities that X = xi for values of yj , where j assumes a value from 1 to m. Similarly, the marginal probabilities for Y are the simple probabilities that Y = yj , where i assumes a value from 1 and n. INDEPENDENT DEDNIM LIKE YOU It is important to note that the distribution satisfies a joint probability function, namely, (a) p(xi , yj ) ≥ 0, (b) n  m  for all i = 1, 2, · · · , n; j = 1, 2, · · · , m. p(xi , yj ) = 1 i=1 j=1 We believe in equality, sustainability and a modern approach to learning. How about you? Apply for a Master’s Programme in Gothenburg, Sweden. PS. Scholarships available for Indian students! Download free eBooks at bookboon.com 12 Click on the ad to read more xn p(x , y ) p(xn , y2 ) ADVANCED TOPICSn IN 1INTRODUCTORY PROBABILITY: A FIRST COURSE IN p(y2 )III Column THEORY p(y1 ) – VOLUME PROBABILITY Totals ··· p(xn , ym ) ··· p(ym ) p(xn ) n  m  PROBABILITY AND DISTRIBUTION FUNCTIONS p(xi , yj ) =OF1 BIVARIATE DISTRIBUTIONS i=1 j=1 Note The marginal probabilities for X are simply the simple probabilities that X = xi for values of yj , where j assumes a value from 1 to m. Similarly, the marginal probabilities for Y are the simple probabilities that Y = yj , where i assumes a value from 1 and n. It is important to note that the distribution satisfies a joint probability function, namely, (a) p(xi , yj ) ≥ 0, (b) n  m  for all i = 1, 2, · · · , n; j = 1, 2, · · · , m. p(xi , yj ) = 1 i=1 j=1 Density and Distribution Functions of Bivariate Distributions Density and Distribution Functions of Bivariate Distributions 9 9 Example 1.1 Example 1.1 (a) For the data in Table 1.2, calculate the joint probabilities of X and Y . (a) For the data in Table 1.2, calculate the joint probabilities of X and Y . (b) Does this distribution satisfy the properties of a joint probability func(b) Does tion? this distribution satisfy the properties of a joint probability function? Solution Solution (a) From Table 1.2, (a) From Table 1.2, element; element; Total number of Total Hencenumber of Hence the cell ({X = 0} ∩ {Y = 1}) = (0, 1) contains one the cell ({X = 0} ∩ {Y = 1}) = (0, 1) contains one elements in all cells is 8. elements in all cells is 8. P ({X = 0} ∩ {Y = 1}) = p(0, 1) P ({X = 0} ∩ {Y = 1}) = p(0, #({X 1) 1 = 0} ∩ {Y = 1}) =1 = n m #({X = 0} ∩ {Y = 1})   = n m = 8 #({X = xi } ∩ {Y = yj }) 8 i j #({X = xi } ∩ {Y = yj }) i j Similarly, Similarly, P ({X P ({X P ({X P ({X P ({X P ({X P ({X P ({X P ({X P ({X = 0} ∩ {Y = 0} ∩ {Y = 0} ∩ {Y = 0} ∩ {Y = 1} ∩ {Y = 1} ∩ {Y = 1} ∩ {Y = 1} ∩ {Y = 1} ∩ {Y = 1} ∩ {Y = 2}) = 2}) = 3}) = 3}) = 1}) = 1}) = 2}) = 2}) = 3}) = 3}) = = = = = = = = = = p(0, 2) = p(0, 2) = p(0, 3) = p(0, 3) = p(1, 1) = p(1, 1) = p(1, 2) = p(1, 2) = p(1, 3) = p(1, 3) = 0 08 80 08 80 08 82 28 81 18 8 =0 =0 =0 =0 =0 =0 1 =1 = 4 41 =1 = 8 8 When probabilities of all possible joint events, P (X = xi , Y = yj ), When probabilities of in allthis possible joint (X probability = xi , Y = ydisj ), have been determined fashion, weevents, have a P joint have been determined in this fashion, we have a joint probability distribution of X and Y and these results may be presented in a two-way tribution of X and Y and results may be presented in a two-way table as shown in the tablethese below: table as shown in the table below: Download free eBooks at bookboon.com 13 8 2 1 P ({X = 1} ∩ {Y = 2}) = p(1, 2) = = 8 4 ADVANCED TOPICS IN INTRODUCTORY 1 1 PROBABILITY: A FIRST COURSE IN AND DISTRIBUTION FUNCTIONS P ({X = 1} ∩ {Y = 3}) = p(1, 3) PROBABILITY = = PROBABILITY THEORY – VOLUME III OF BIVARIATE DISTRIBUTIONS 8 8 10 10 When probabilities of all possible joint events, P (X = xi , Y = yj ), Topics in Introductory Probability have been determined inAdvanced this fashion, we have a joint probability distribution of X and Y andAdvanced these results mayinbe presented inProbability a two-way Topics Introductory table as shown in the table below: X Y 1 3 X Y2 0 1/8 02 03 1 10 1/8 0 2/8 1/8 0 0 21 00 2/8 2/8 1/8 1/8 32 1/8 0 0 0 2/8 1/8 3 1/8 0 0 (b) From the table above, (b) From the, table above,for all i = 0, 1, 2, 3; j = 1, 2, 3. (i) p(x i yj ) ≥ 0, 3  3 (i)  p(x for i , yj ) ≥ 0, 1 all1 i =2 0, 1,22, 3;1 j =1 1, 2, 3. (ii) p(xi , yj ) = + + + + + = 1 3 3 81 81 82 82 81 81   i=0 j=1 (ii) p(xi , yj ) = + + + + + = 1 8 8 8 8 8 8 j=1 distribution is a joint probability function. Hencei=0 this Hence this distribution is a joint probability function. Joint Probability Distribution of Bivariate Random Variables in Expression Form Joint Probability Distribution of Bivariate Random Variables in Expression Form Sometimes the joint probability distribution of discrete random variables X and Y is given by a formula. Sometimes the joint probability distribution of discrete random variables X and Y is given by a formula. Example 1.2 Given the1.2 function Example Given the function p(x, y) = k(3x + 2y), x = 0, 1; y = 0, 1, 2 p(x, y) = k(3x + 2y), x = 0, 1; y = 0, 1, 2 (a) Find the constant k > 0 such that the p(x, y) is a joint probability (a) mass Find function. the constant k > 0 such that the p(x, y) is a joint probability mass function. (b) Present it in a tabular form for the probabilities associated with the y). Obtain the the rowprobabilities and column totals. (b) sample Present points it in a(x, tabular form for associated with the sample points (x, y). Obtain the row and column totals. Solution (a) (i) p(x, y) ≥ 0 Solution (a) (i) and p(x, y) Density Functions  of1 Bivariate Distributions 2 ≥0 1 Distribution   Density and Distribution Functions of Bivariate Distributions (ii) k(3x + 2y) = k [(3x + 0) + (3x + 2) + (3x + 4)] (ii) 2 1   x=0 y=0 x=0 y=0 k(3x + 2y) = = = = = = = 1  x=0 1 k  [(3x + 0) 1 (9x + 6)  k x=0 k x=0(9x + 6) x=0 11 11 + (3x + 2) + (3x + 4)] k[(0 + 6) + (9 + 6)] k[(0 + 6) + (9 + 6)] 21k 21k For p(x, y) to be a joint probability function we must have 21k = 1 from For p(x, y) to be a joint probability function we must have 21k = 1 from which which 1 k= 1 k = 21 21 (b) For the sample point {X = 0, Y = 0} = (0, 0) (b) For the sample point {X = 0, Y = 0} = (0, 0) Download free eBooks at bookboon.com 1 p(0, 0) = 1 [3(0) + 2(0)] = 0 14= 0 p(0, 0) = 21 [...
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