DOCSLIB.ORG

Data Analysis for Two-Way Tables Section 9.1: Chi-Square Test for Two-Way Tables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Data Analysis for Two-Way Tables Section 9.1: Chi-Square Test for Two-Way Tables Section 2.5: Data Analysis for Two-Way Tables Section 9.1: Chi-square test for Two-Way Tables Learning goals for this chapter: Find the joint, marginal, and conditional distributions from a two-way table of the counts by hand and with SPSS. Determine from the wording of the story whether the question is asking for a joint, marginal, or conditional percentage/probability. Know when it two-way tables and the chi-square test are the correct statistical technique for a story. Perform a hypothesis test for a 2 test, including: stating the hypotheses, obtaining the test statistic and P-value from SPSS, and writing a conclusion in terms of the story. Check assumption to see if it is appropriate to use a test using the footnote of the SPSS test. Two-way tables and the chi-square test are used when you are studying the association between 2 categorical variables. cell # The joint distribution of the 2 categorical variables is the (the inner squares). total # All the joint distribution should add to 1. The marginal distribution allows us to study 1 variable at a time. You get them just by adding across a row or down a column for the specific variable you are interested in. The marginals are written in the margins of the table (far right and very bottom). The marginals for the row variable should add to 1. The marginals for the column variable should add to 1. Conditional distribution: If you know one variable for sure (you have “reduced your world”), what are the respective percentages for the other variable? Bar graphs are a good way to demonstrate conditional distributions. Hypothesis testing with 2-way tables H0: There is no association between the row and column variables in the population. Ha: There is an association between the row and column variables in the population. To test the null hypothesis, compare observed cell counts with expected cell counts calculated under the assumption that the null hypothesis is true. 1 Test statistic: Chi Square Test Statistic observed count - expected count 2 X 2 expected count row total x column total Expected count = , n where n = total # of observations for the table. The X2 test statistic has an approximately chi-square distribution. To use the chi-square table, you need the degrees of freedom, (r-1)(c-1). Go to Table F in the back of the book. WE WILL LET SPSS CALCULATE THE TEST STATISTIC AND P-VALUE FOR US. YOU DO NOT NEED TO KNOW HOW TO USE THE TABLE. P-value for chi-square test is: PX()22 (We’ll be using SPSS to do the test.) The chi-square test becomes more accurate as the cell counts increase and for tables larger than 2x2. For tables larger than 2x2: use chi-square test whenever the average of the expected counts is 5 or more and the smallest expected count is 1 or more <20% of cells have expected counts of less than 5. For 2x2 tables: use chi-square test whenever all 4 expected cell counts to be 5 or more Example: Market researchers know that background music can influence the mood and purchasing behavior of customers. One study in a supermarket in Northern Ireland compared 3 treatments: no music, French accordion music, and Italian string music. Under each condition, the researchers recorded the number of bottles of French, Italian, and other wine purchased. Here is the 2-way table that summarizes the data in counts (total # of bottles sold = 243: Music Wine None French Italian French 30 39 30 Italian 11 1 19 Other 43 35 35 2 Calculate the joint distribution for music and wine: Music Wine None French Italian French 12.3 16.0 12.3 Italian 4.5 0.4 7.8 Other 17.7 14.4 14.4 Calculate the marginal distribution for music: Music Wine None French Italian French 12.3 16.0 12.3 Italian 4.5 0.4 7.8 Other 17.7 14.4 14.4 Marg. for music 34.6 30.9 34.6 Calculate the marginal distribution for wine: Music Wine None French Italian Marg. for wine French 12.3 16.0 12.3 40.7 Italian 4.5 0.4 7.8 12.8 Other 17.7 14.4 14.4 46.5 Marg. for music 34.6 30.9 34.6 100 3 Questions (joint, marginal, conditional?): 1. “What percent of all wine bought was Italian with French music playing in the store?” 2. “Of the Italian wine purchased, what percent was from a store playing French music?” 3. “What percent of wine bought was Italian?” 4. “What percent of the wine purchased from French music-playing stores was French?” 5. “What percent of wine was purchased from a store with no music playing?” Using SPSS, set up the data so that you have a wine column, a music column, and a purchase column (where you will input the counts inside the chart). Wine Music Purchase French None 30 Italian None 11 Other None 43 French French 39 Italian French 1 Other French 35 French Italian 30 Italian Italian 19 Other Italian 35 Then go to Data --> Weight Cases. Click “Weight cases by” and then move “purchase” into the “frequency variable” box. Click OK. Do Analyze--> Descriptive Statistics --> Crosstabs. Make sure “observed” is checked. Put “wine” into the “Rows” box and “music” into the “Columns” box. Click OK. You will get: 4 Type of Wine * Type of Music Crosstabulation Count Type of Music French Italian None Total Type of French 39 30 30 99 Wine Italian 1 19 11 31 Other 35 35 43 113 Total 75 84 84 243 Then if you want the %s for joint and marginal distributions instead of counts, you go back to your data and do Analyze --> Descriptive Statistics --> Crosstabs --> (your rows and columns should still be entered from the previous step) --> Click “Cells” --> Click “Total.” Also, un-click “observed” so your table won’t also include the counts and be too crowded. Click “Continue” and then “OK.” You will get: Type of Wine * Type of Music Crosstabulation % of Total Type of Music French Italian None Total Type of French 16.0% 12.3% 12.3% 40.7% Wine Italian .4% 7.8% 4.5% 12.8% Other 14.4% 14.4% 17.7% 46.5% Total 30.9% 34.6% 34.6% 100.0% Is there a relationship in the population between the type of wine purchased and the type of music that is playing? Perform a significance test, and write a short summary of your conclusion. Hypotheses: Test statistic: P-value: Conclusion in terms of the story: Was it appropriate to use the chi-square test here? Justify your answer. 5 To make SPSS do the hypothesis test, you go back to Analyze --> Descriptive Statistics -- > Crosstabs --> Cells. Then click “total” to make their checks go away. Also click “expected” under “counts.” Click Continue. Then click Statistics --> Chi-Square --> Continue --> OK. You will get: Chi-Square Tests Asymp. Sig. Value df (2-sided) Pearson Chi-Square 18.279a 4 .001 Likelihood Ratio 21.875 4 .000 N of Valid Cases 243 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 9.57. Use the “Pearson Chi-Square” to get your X2 test statistic, and the “Asymp. Sig.” to get the P-value. Example: Psychological and social factors can influence the survival of patients with serious diseases. One study examined the relationship between survival of patients with coronary heart disease and pet ownership. Each of 92 patients was classified as having a pet or not and by whether they survived for one year. The researchers suspect that having a pet might be connected to the patient status. Here are the data: Pet ownership Patient Status No Yes Alive 28 50 Dead 11 3 Total 39 53 a) Find the joint and marginal distributions (in probabilities) of patient status and pet ownership. Pet ownership Patient Status No Yes Marg, for status Alive 0.304 0.543 0.847 Dead 0.120 0.033 0.153 Marg. for pets 0.424 0.576 b) Assuming a patient is still alive, what is the probability he owns a pet? Is this a joint, marginal, or conditional probability? 6 c) What is the probability a patient is still alive and owns a pet? Is this a joint, marginal, or conditional probability? d) What is the probability a patient owns a pet? Is this a joint, marginal, or conditional probability? e) State the hypotheses for a 2 test of this problem, find the X2 test statistic, its degrees of freedom, and the P-value. State your conclusion in terms of the original problem. Hypotheses: Test statistic: P-value: Conclusion in terms of the story: Chi-Square Tests Asymp. Sig. Exact Sig. Exact Sig. Value df (2-sided) (2-sided) (1-sided) Pearson Chi-Square 8.851(b) 1 .003 Continuity 7.190 1 .007 Correction(a) Likelihood Ratio 9.011 1 .003 Fisher's Exact Test .006 .004 Linear-by-Linear Association 8.755 1 .003 N of Valid Cases 92 7 Student Handout for M&Ms/Skittles Activity (Chapter 9: Two Way Distributions) Part 1: Plain vs. Peanut M&Ms 1. Your data for plain (mine for peanut), in counts: Brown Yellow Red Blue Orange Green Total Plain Peanut 2 3 5 0 8 4 22 Total Overall total number of plain and peanut M&Ms counted: 2. Joint Distribution (in white boxes). Divide each count above by the overall total of M&Ms.
Load more

Recommended publications
  • Chapter 6 Continuous Random Variables and Probability
    EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2019 Chapter 6 Continuous Random Variables and Probability Distributions Chap 6-1 Probability Distributions Probability Distributions Ch. 5 Discrete Continuous Ch. 6 Probability Probability Distributions Distributions Binomial Uniform Hypergeometric Normal Poisson Exponential Chap 6-2/62 Continuous Probability Distributions § A continuous random variable is a variable that can assume any value in an interval § thickness of an item § time required to complete a task § temperature of a solution § height in inches § These can potentially take on any value, depending only on the ability to measure accurately. Chap 6-3/62 Cumulative Distribution Function § The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x F(x) = P(X £ x) § Let a and b be two possible values of X, with a < b. The probability that X lies between a and b is P(a < X < b) = F(b) -F(a) Chap 6-4/62 Probability Density Function The probability density function, f(x), of random variable X has the following properties: 1. f(x) > 0 for all values of x 2. The area under the probability density function f(x) over all values of the random variable X is equal to 1.0 3. The probability that X lies between two values is the area under the density function graph between the two values 4. The cumulative density function F(x0) is the area under the probability density function f(x) from the minimum x value up to x0 x0 f(x ) = f(x)dx 0 ò xm where
    [Show full text]
  • Extremal Dependence Concepts
    Extremal dependence concepts Giovanni Puccetti1 and Ruodu Wang2 1Department of Economics, Management and Quantitative Methods, University of Milano, 20122 Milano, Italy 2Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada Journal version published in Statistical Science, 2015, Vol. 30, No. 4, 485{517 Minor corrections made in May and June 2020 Abstract The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements.
    [Show full text]
  • Joint Probability Distributions
    ST 380 Probability and Statistics for the Physical Sciences Joint Probability Distributions In many experiments, two or more random variables have values that are determined by the outcome of the experiment. For example, the binomial experiment is a sequence of trials, each of which results in success or failure. If ( 1 if the i th trial is a success Xi = 0 otherwise; then X1; X2;:::; Xn are all random variables defined on the whole experiment. 1 / 15 Joint Probability Distributions Introduction ST 380 Probability and Statistics for the Physical Sciences To calculate probabilities involving two random variables X and Y such as P(X > 0 and Y ≤ 0); we need the joint distribution of X and Y . The way we represent the joint distribution depends on whether the random variables are discrete or continuous. 2 / 15 Joint Probability Distributions Introduction ST 380 Probability and Statistics for the Physical Sciences Two Discrete Random Variables If X and Y are discrete, with ranges RX and RY , respectively, the joint probability mass function is p(x; y) = P(X = x and Y = y); x 2 RX ; y 2 RY : Then a probability like P(X > 0 and Y ≤ 0) is just X X p(x; y): x2RX :x>0 y2RY :y≤0 3 / 15 Joint Probability Distributions Two Discrete Random Variables ST 380 Probability and Statistics for the Physical Sciences Marginal Distribution To find the probability of an event defined only by X , we need the marginal pmf of X : X pX (x) = P(X = x) = p(x; y); x 2 RX : y2RY Similarly the marginal pmf of Y is X pY (y) = P(Y = y) = p(x; y); y 2 RY : x2RX 4 / 15 Joint
    [Show full text]
  • (Introduction to Probability at an Advanced Level) - All Lecture Notes
    Fall 2018 Statistics 201A (Introduction to Probability at an advanced level) - All Lecture Notes Aditya Guntuboyina August 15, 2020 Contents 0.1 Sample spaces, Events, Probability.................................5 0.2 Conditional Probability and Independence.............................6 0.3 Random Variables..........................................7 1 Random Variables, Expectation and Variance8 1.1 Expectations of Random Variables.................................9 1.2 Variance................................................ 10 2 Independence of Random Variables 11 3 Common Distributions 11 3.1 Ber(p) Distribution......................................... 11 3.2 Bin(n; p) Distribution........................................ 11 3.3 Poisson Distribution......................................... 12 4 Covariance, Correlation and Regression 14 5 Correlation and Regression 16 6 Back to Common Distributions 16 6.1 Geometric Distribution........................................ 16 6.2 Negative Binomial Distribution................................... 17 7 Continuous Distributions 17 7.1 Normal or Gaussian Distribution.................................. 17 1 7.2 Uniform Distribution......................................... 18 7.3 The Exponential Density...................................... 18 7.4 The Gamma Density......................................... 18 8 Variable Transformations 19 9 Distribution Functions and the Quantile Transform 20 10 Joint Densities 22 11 Joint Densities under Transformations 23 11.1 Detour to Convolutions......................................
    [Show full text]
  • Probability for Statistics: the Bare Minimum Prerequisites
    Probability for Statistics: The Bare Minimum Prerequisites This note seeks to cover the very bare minimum of topics that students should be familiar with before starting the Probability for Statistics module. Most of you will have studied classical and/or axiomatic probability before, but given the various backgrounds it's worth being concrete on what students should know going in. Real Analysis Students should be familiar with the main ideas from real analysis, and at the very least be familiar with notions of • Limits of sequences of real numbers; • Limit Inferior (lim inf) and Limit Superior (lim sup) and their connection to limits; • Convergence of sequences of real-valued vectors; • Functions; injectivity and surjectivity; continuous and discontinuous functions; continuity from the left and right; Convex functions. • The notion of open and closed subsets of real numbers. For those wanting for a good self-study textbook for any of these topics I would recommend: Mathematical Analysis, by Malik and Arora. Complex Analysis We will need some tools from complex analysis when we study characteristic functions and subsequently the central limit theorem. Students should be familiar with the following concepts: • Imaginary numbers, complex numbers, complex conjugates and properties. • De Moivre's theorem. • Limits and convergence of sequences of complex numbers. • Fourier transforms and inverse Fourier transforms For those wanting a good self-study textbook for these topics I recommend: Introduction to Complex Analysis by H. Priestley. Classical Probability and Combinatorics Many students will have already attended a course in classical probability theory beforehand, and will be very familiar with these basic concepts.
    [Show full text]
  • Using Learned Conditional Distributions As Edit Distance Jose Oncina, Marc Sebban
    Using Learned Conditional Distributions as Edit Distance Jose Oncina, Marc Sebban To cite this version: Jose Oncina, Marc Sebban. Using Learned Conditional Distributions as Edit Distance. Structural, Syntactic, and Statistical Pattern Recognition, Joint IAPR International Workshops, SSPR 2006 and SPR 2006, Aug 2006, Hong Kong, China. pp 403-411, ISBN 3-540-37236-9. hal-00322429 HAL Id: hal-00322429 https://hal.archives-ouvertes.fr/hal-00322429 Submitted on 18 Sep 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Using Learned Conditional Distributions as Edit Distance⋆ Jose Oncina1 and Marc Sebban2 1 Dep. de Lenguajes y Sistemas Informaticos,´ Universidad de Alicante (Spain) [email protected] 2 EURISE, Universite´ de Saint-Etienne, (France) [email protected] Abstract. In order to achieve pattern recognition tasks, we aim at learning an unbiased stochastic edit distance, in the form of a finite-state transducer, from a corpus of (input,output) pairs of strings. Contrary to the state of the art methods, we learn a transducer independently on the marginal probability distribution of the input strings. Such an unbiased way to proceed requires to optimize the pa- rameters of a conditional transducer instead of a joint one.
    [Show full text]
  • Multivariate Rank Statistics for Testing Randomness Concerning Some Marginal Distributions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MULTIVARIATE ANALYSIS 5, 487-496 (1975) Multivariate Rank Statistics for Testing Randomness Concerning Some Marginal Distributions MARIE HuS~ovli Department of Statistics, CharZes University, Prague, Czechoskwahia Communicated by P. K. Sen In the present paper there is proposed a rank statistic for multivariate testing of randomness concerning some marginal distributions. The asymptotic distribu- tion of this statistic under hypothesis and “near” alternatives is treated. 1. INTRODUCTION Let Xj = (X, ,..., XJ, j = I,..., IV, be independent p-dimensional random variables and letF,&, y) andFdj(x) denote the distribution function of (Xii , XV!) and X,, , respectively. Fi, is assumedcontinuous throughout the whole paper, i=l ,..., p,j = l,...) N. We are interested in testing hypothesis of randomness of the univariate marginal distributions: for all x, i = l,..., p, j = 1,. N (1.1) against for all x, i = I,..., p, j = I ,..., N, (1.2) where $,, are unknown parameters. Or more generally, we are interested in testing hypothesis of randomnessassociated with some marginal distributions: F,“(x”) = F”(X”), u = l,..., Y, j = l,..., N, (1.3) against F,y(X”) = qx”; El;), Y = l,..., r, j = I,..., N, (1.4) Received September 1974; revised April 15, 1975. AMS subject classifications: Primary 62G10, Secondary 62HlO. Key words and phrases: Rank statistics; Testing randomness, Asymptotic distribution. 487 Copyright 0 1975 by Academic Press, Inc. All rights of reproduction in any form reserved. 488 MARIE HU?,KOVsi where FjV(xy) is the marginal distribution of the subvector Xy, Y = I,..., r, Xl ,***>x7, is a partition of the vector x, i.e., x’ = (xl’,..., xr’) and Oj’ = (0,” ,..., 0;‘) is a vector of unknown parameters.
    [Show full text]
  • Random Variables and Probability Distributions
    Schaum's Outline of Probability and Statistics CHAPTERCHAPTER 122 Random Variables and Probability Distributions Random Variables Suppose that to each point of a sample space we assign a number. We then have a function defined on the sam- ple space. This function is called a random variable (or stochastic variable) or more precisely a random func- tion (stochastic function). It is usually denoted by a capital letter such as X or Y. In general, a random variable has some specified physical, geometrical, or other significance. EXAMPLE 2.1 Suppose that a coin is tossed twice so that the sample space is S ϭ {HH, HT, TH, TT}. Let X represent the number of heads that can come up. With each sample point we can associate a number for X as shown in Table 2-1. Thus, for example, in the case of HH (i.e., 2 heads), X ϭ 2 while for TH (1 head), X ϭ 1. It follows that X is a random variable. Table 2-1 Sample Point HH HT TH TT X 2110 It should be noted that many other random variables could also be defined on this sample space, for example, the square of the number of heads or the number of heads minus the number of tails. A random variable that takes on a finite or countably infinite number of values (see page 4) is called a dis- crete random variable while one which takes on a noncountably infinite number of values is called a nondiscrete random variable. Discrete Probability Distributions Let X be a discrete random variable, and suppose that the possible values that it can assume are given by x1, x2, x3, .
    [Show full text]
  • Multivariate Distributions
    IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Multivariate Distributions We will study multivariate distributions in these notes, focusing1 in particular on multivariate normal, normal-mixture, spherical and elliptical distributions. In addition to studying their properties, we will also discuss techniques for simulating and, very briefly, estimating these distributions. Familiarity with these important classes of multivariate distributions is important for many aspects of risk management. We will defer the study of copulas until later in the course. 1 Preliminary Definitions Let X = (X1;:::Xn) be an n-dimensional vector of random variables. We have the following definitions and statements. > n Definition 1 (Joint CDF) For all x = (x1; : : : ; xn) 2 R , the joint cumulative distribution function (CDF) of X satisfies FX(x) = FX(x1; : : : ; xn) = P (X1 ≤ x1;:::;Xn ≤ xn): Definition 2 (Marginal CDF) For a fixed i, the marginal CDF of Xi satisfies FXi (xi) = FX(1;:::; 1; xi; 1;::: 1): It is straightforward to generalize the previous definition to joint marginal distributions. For example, the joint marginal distribution of Xi and Xj satisfies Fij(xi; xj) = FX(1;:::; 1; xi; 1;:::; 1; xj; 1;::: 1). If the joint CDF is absolutely continuous, then it has an associated probability density function (PDF) so that Z x1 Z xn FX(x1; : : : ; xn) = ··· f(u1; : : : ; un) du1 : : : dun: −∞ −∞ Similar statements also apply to the marginal CDF's. A collection of random variables is independent if the joint CDF (or PDF if it exists) can be factored into the product of the marginal CDFs (or PDFs). If > > X1 = (X1;:::;Xk) and X2 = (Xk+1;:::;Xn) is a partition of X then the conditional CDF satisfies FX2jX1 (x2jx1) = P (X2 ≤ x2jX1 = x1): If X has a PDF, f(·), then it satisfies Z xk+1 Z xn f(x1; : : : ; xk; uk+1; : : : ; un) FX2jX1 (x2jx1) = ··· duk+1 : : : dun −∞ −∞ fX1 (x1) where fX1 (·) is the joint marginal PDF of X1.
    [Show full text]
  • Comparing Conditional and Marginal Direct Estimation of Subgroup Distributions
    RESEARCH REPORT January 2003 RR-03-02 Comparing Conditional and Marginal Direct Estimation of Subgroup Distributions Matthias von Davier Research & Developm ent Division Princeton, NJ 08541 Comparing Conditional and Marginal Direct Estimation of Subgroup Distributions Matthias von Davier Educational Testing Service, Princeton, NJ January 2003 Research Reports provide preliminary and limited dissemination of ETS research prior to publication. They are available without charge from: Research Publications Office Mail Stop 10-R Educational Testing Service Princeton, NJ 08541 Abstract Many large-scale assessment programs in education utilize “conditioning models” that incorporate both cognitive item responses and additional respondent background variables relevant for the population of interest. The set of respondent background variables serves as a predictor for the latent traits (proficiencies/abilities) and is used to obtain a conditional prior distribution for these traits. This is done by estimating a linear regression, assuming normality of the conditional trait distributions given the set of background variables. Multiple imputations, or plausible values, of trait parameter estimates are used in addition to or, better, on top of the conditioning model—as a computationally convenient approach to generating consistent estimates of the trait distribution characteristics for subgroups in complex assessments. This report compares, on the basis of simulated and real data, the conditioning method with a recently proposed method of estimating subgroup distribution statistics that assumes marginal normality. Study I presents simulated data examples where the marginal normality assumption leads to a model that produces appropriate estimates only if subgroup differences are small. In the presence of larger subgroup differences that cannot be fitted by the marginal normality assumption, however, the proposed method produces subgroup mean and variance estimates that differ strongly from the true values.
    [Show full text]
  • Bivariate Kumaraswamy Models Via Modified Symmetric FGM Copulas: Properties and Applications in Insurance Modeling
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 September 2017 doi:10.20944/preprints201709.0115.v1 Peer-reviewed version available at J. Risk Financial Manag. 2017, 10, 19; doi:10.3390/jrfm10040019 Bivariate Kumaraswamy models via modified symmetric FGM copulas: Properties and Applications in Insurance modeling Indranil Ghosh Department of Mathematics and Statistics, University of North Carolina, Wilmington, USA Abstract A copula is a useful tool for constructing bivariate and/or multivariate dis- tributions. In this article, we consider a new modified class of (Farlie-Gumbel- Morgenstern) FGM bivariate copula for constructing several different bivariate Ku- maraswamy type copulas and discuss their structural properties, including depen- dence structures. It is established that construction of bivariate distributions by this method allows for greater flexibility in the values of Spearman's correlation coefficient, ρ and Kendall's τ. For illustrative purposes, one representative data set is utilized to exhibit the applicability of these proposed bivariate copula models. Key Words: Bivariate Kumaraswamy distribution, Copula based construction, Kendall's tau, Dependence structures 1 Introduction Over the last decade or so, there has been a growing interest in constructing various bivariate distributions and study it's dependence structure. For an excellent survey on this, an interested reader is suggested to see Balakrishnan et al. (2009) and the references therein. Of late, copula based methods of construction have also gained a considerable amount of attention, mainly due to it's analytical tractability in the sense of discussing 1 © 2017 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 September 2017 doi:10.20944/preprints201709.0115.v1 Peer-reviewed version available at J.
    [Show full text]
  • Chapter 2 Random Variables and Distributions
    Chapter 2 Random Variables and Distributions CHAPTER OUTLINE Section 1 Random Variables Section 2 Distributions of Random Variables Section 3 Discrete Distributions Section 4 Continuous Distributions Section 5 Cumulative Distribution Functions Section 6 One-Dimensional Change of Variable Section 7 Joint Distributions Section 8 Conditioning and Independence Section 9 Multidimensional Change of Variable Section 10 Simulating Probability Distributions Section 11 Further Proofs (Advanced) In Chapter 1, we discussed the probability model as the central object of study in the theory of probability. This required defining a probability measure P on a class of subsets of the sample space S. It turns out that there are simpler ways of presenting a particular probability assignment than this — ways that are much more convenient to work with than P. This chapter is concerned with the definitions of random variables, distribution functions, probability functions, density functions, and the development of the concepts necessary for carrying out calculations for a probability model using these entities. This chapter also discusses the concept of the conditional distribution of one random variable, given the values of others. Conditional distributions of random variables provide the framework for discussing what it means to say that variables are related, which is important in many applications of probability and statistics. 33 34 Section 2.1: Random Variables 2.1 Random Variables The previous chapter explained how to construct probability models, including a sam- ple space S and a probability measure P. Once we have a probability model, we may define random variables for that probability model. Intuitively, a random variable assigns a numerical value to each possible outcome in the sample space.
    [Show full text]