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A General Approach to Teaching Random Variables 1 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 A GENERAL APPROACH TO TEACHING RANDOM VARIABLES Farida Kachapova School of Computing and Mathematical Sciences Auckland University of Technology Private Bag 92006 Auckland 1142 New Zealand Email: [email protected] Abstract The paper describes a teaching approach in post-calculus probability courses when initial topics on random variables are taught in general form instead of doing it separately for discrete and continuous cases. Some common student misconceptions about random variables are analyzed and accompanied with counter-examples. Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching- Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. www.hostos.cuny.edu/departments/math/mtrj 2 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 INTRODUCTION Most research on teaching probability focus on basic probability and statistical applications. The teaching of probability concepts, including random variables, is studied by Larsen (2006), Miller (1998), Leviatan (2003), Ballman (1997), and Hernandez, Albert Huerta and Batanero (2006). Much research has been done on students’ thinking about probability and related misconceptions (see, for example, Ancker, 2006; Rossman, 1995; San Martin, 2006; Warner, Pendergraft, and Webb, 1998). Random variable is “... a fundamental stochastic idea” and “... is the support of many probability and statistics subjects” (Hernandez, Albert Huerta and Batanero, 2006). So it is important that the students in probability courses gain a deep understanding of the concept of random variable, which starts with a proper definition. In post-calculus university courses the students have sufficient mathematical maturity to learn correct definitions and derive mathematical theorems from them. As Leviatan (2003) wrote, “The formal axiomatic approach to probability is important as they are the basis for all other probabilistic rules.” The axiomatic foundation developed by Kolmogorov in 1931 (see the English translation: Kolmogorov, 1950). In Section 2 we look at some common students’ misconceptions about events and random variables. In Section 3 we make suggestions about teaching random variables. In most probability courses random variables are taught in two separate chapters containing the material on discrete variables and continuous ones, respectively. So many topics are taught twice. The expectations of discrete and continuous random variables are defined separately, so the students perceive them as two different concepts and miss the general mathematical meaning of the expected value. Besides some theorems that are true for arbitrary random variables, are stated only for the discrete and continuous ones. So instead of losing generality we suggest to use Riemann-Stieltjes integral to introduce the Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching- Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. www.hostos.cuny.edu/departments/math/mtrj 3 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 expectation of a random variable in general form and to develop the theory for arbitrary random variables as long as possible before teaching particular properties of the discrete and continuous variables. This makes the course slightly more advanced, so our suggestions are more appropriate for the university courses in quantitative areas. Our suggestions do not target introductory service courses in statistics, where students have a limited mathematical background and common teaching methods with minimum mathematics are appropriate. 2. COMMON MISCONCEPTIONS 2.1. EVENTS One of the common misconceptions in probability is the belief that an event is an arbitrary subset of a sample space, which is usually correct in the case of a finite sample space but not in the general case; sometimes Venn diagrams can strengthen this misconception. The misconception can be avoided if we use the axiomatic approach to probability where a subset A of a sample space is considered an event if its probability measure P(A) exists (that is an event is a measurable subset of ). All events on the sample space make a -field. Thus, in probability theory every experiment is assigned a probability space (, A, P), where is a sample space, A is a -field and P is a probability measure; then an event is defined as any element of A. Non-measurable sets are not events, since they cannot be assigned probabilities. Whether non-measurable sets exist, appears to be a complicated question, and the probability students can be given a brief overview of the topic: 1) an example of non-Borel set on was constructed by Luzin (1953); 2) Vitali (1905) proved the existence of a set of real numbers that is not Lebesgue measurable, with the assumption of the axiom of choice; Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching- Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. www.hostos.cuny.edu/departments/math/mtrj 4 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 3) Solovay (1970) created a model, where all subsets of are Lebesgue measurable and the axiom of choice is false; this demonstrates that the existence of non-measurable subsets of cannot be proven without the axiom of choice. The standard approach in mathematics is to accept the existence of non- measurable sets; therefore in probability theory some sample spaces can have subsets, which are not events. 2.2. RANDOM VARIABLE Random variable is a fundamental concept in probability theory that is not intuitive, so some students develop misconceptions about it. One common misconception is that a random variable X is any real-valued function on the sample space. Students can overcome this misconception by learning the most important characteristic of the random variable – its distribution function FX : FX (x) = P(X x). For the distribution function to exist the probability P(X x) should be defined for any real number x. Therefore X is a valid random variable only if for any xR the probability measure P(X x) is defined, that is {X x} is an event. Mathematically a random variable on the probability space (, A, P) is defined as a function X: with the property: (x) ({ X x }A). (1) The condition (1) is essential because the existence of non-measurable sets is accepted in mathematics, as discussed before. 2.1 CONTINUOUS VARIABLE Some students’ misconceptions in probability relate to continuous random variables, which make a harder topic than discrete random variables. One of the misconceptions is defining a continuous random variable as a variable with a non-countable set of values Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching- Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. www.hostos.cuny.edu/departments/math/mtrj 5 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 (sometimes this set is expected to be an interval or a combination of intervals). This leads to the misconception that any random variable is either discrete or continuous. This misconception might be related to statistics courses where numerical data are categorized into two types: discrete and continuous. To help students to overcome these misconceptions we can emphasize the following standard definition. A random variable X is continuous if its distribution function F can be x expressed as F (x) = f tdt for some integrable non-negative function f , which is called the density function of X. A simple example of a mixed random variable can clarify the matter further. Example 1. Mixed random variable. Consider the following function: 0 if x 0, 0.2x if 0 x 1, Fx 0.2x 0.6 if 1 x 2, 1 if x 2. u 1 F(x) 0.8 0.2 Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching- Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. 0 www.hostos.cuny.edu/departments/math/mtrj1 2 x 6 MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 2 May 2012 Clearly F is non-decreasing, right-continuous and has limits 0 and 1 at and , respectively. So F is the distribution function of some random variable X. Here P (X = 1) = 0.6. If we assume that X has a density function f (x), then f (x) = F (x) for any real x except points 0, 1 and 2; so 0.6 = P (X = 1) F(1) = 1 1 f xdx 0.2dx 0.2, which is a contradiction.
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