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ECONOMETRICS I APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS Textbook: Damodar N. Gujarati (2004) Basic Econometrics, 4th edition, The McGraw-Hill Companies A.1: SUMMATION AND PRODUCT OPERATORS A.1: SUMMATION AND PRODUCT OPERATORS A.2 SAMPLE SPACE, SAMPLE POINTS AND EVENTS • The set of all possible outcomes of a random, or chance, experiment is called the population, or sample space, and each member of this sample space is called a sample point. • Example: Remember tossing a coin… head, tail… the sample space consists of these four possible outcomes: HH, HT, TH, and TT. • An event is a subset of the sample space. A.2 SAMPLE SPACE, SAMPLE POINTS AND EVENTS • An event is a subset of the sample space. • Events are said to be mutually exclusive if the occurrence of one event precludes the occurrence of another event. • Events are said to be (collectively) exhaustible if they exhaust all the possible outcomes of an experiment. • Thus, in the example, the events (a) two heads, (b) two tails, and (c) one tail, one head exhaust all the outcomes; hence they are (collectively) exhaustive events. A.3 PROBABILITY AND RANDOM VARIABLES • Let A be an event in a sample space. By P(A), the probability of the event A, we mean the proportion of times the event A will occur in repeated trials of an experiment. A.3 PROBABILITY AND RANDOM VARIABLES • A variable whose value is determined by the outcome of a chance experiment is called a random variable (rv). Random variables are usually denoted by the capital letters X, Y, Z, and so on, and the values taken by them are denoted by small letters x, y, z, and so on. A random variable may be either discrete or continuous. A.3 PROBABILITY AND RANDOM VARIABLES • A discrete rv takes on only a finite (or countably infinite) number of values. For example, in throwing two dice, each numbered 1 to 6, if we define the random variable X as the sum of the numbers showing on the dice, then X will take one of these values: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12. Hence it is a discrete random variable. A.3 PROBABILITY AND RANDOM VARIABLES • A continuous rv, on the other hand, is one that can take on any value in some interval of values.. The height of an individual is a continuous variable—in the range, say, 60 to 65 inches it can take any value, depending on the precision of measurement. A.4 PROBABILITY DENSITY FUNCTION (PDF) • Probability Density Function of a Discrete Random Variable A.4 PROBABILITY DENSITY FUNCTION (PDF) • Probability Density Function of a Continuous Random Variable A.4 PROBABILITY DENSITY FUNCTION (PDF) • Probability Density Function of a Continuous Random Variable (Figure A.2) P(a ≤ x ≤ b) means the probability that X lies in the interval a to b. A.4 PROBABILITY DENSITY FUNCTION (PDF) • Joint Probability Density Functions A.4 PROBABILITY DENSITY FUNCTION (PDF) • Joint Probability Density Functions A.5 CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS – Expected Value A.5 CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS – Variance A.5 CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS – Covariance A.6 SOME IMPORTANT THEORETICAL PROBABILITY DISTRIBUTIONS • • • • Normal distribution (Z) The χ2 (Chi-Square) Distribution Student’s t Distribution The F Distribution