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The Summation Notation • The sum of a large number of terms occurs frequently in econometrics. There is an abbreviated notation for such sums. The upper case Greek letter (sigma) is used to indicate a summation and the terms are generally indexed by subscripts. 4 The symbol X i read ``the sum of X sub i as i goes from i1 1 to 4,'' represents the sum X1 + X 2 + X 3 + X 4 . Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Examples 5 Example 1: X i 1 i X1 + X 2 + X 3 X 4 X 5 7 Example 2: (2k -1) =49 k=1 3 Example 3: ( X i 3) 2 ( X 1 3) 2 ( X 2 3) 2 ( X 3 3) 2 i 1 2 Example 4: ( X i 3) X 1 X 2 X 3 X 4 X 5 15 5 2 i 1 2 Example 5: (X i 1 i 3)(Yi 5) ( X 1 3)(Y1 5) ( X 2 3)(Y2 5) Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Example • If five people are asked their ages the results might be summarized in the table Person 1 2 3 4 5 Age (X) 23 19 40 22 35 and the sum of their ages can be represented as 23 + 19 + 40 + 22 + 36 = 140 Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Properties of the Summation Operator n Property 1: If k is a constant k nk i 1 n n Property 2: kX i 1 i k Xi i 1 X Yi n n n Property 3: i 1 i X Y i 1 i i 1 i a bX na b X ; a and b are constants n n Property 4: i i i 1 i 1 Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Sample Average Sample Average (or sample mean): n 1 X Xi n i 1 n X i 1 i n X Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Results n 1. If di X i X then show that di 0 i=1 X X X n X n n 2 2 2 2. i i i=1 i=1 X X Yi Y X i Yi n X Y n n 3. i i=1 i=1 Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Example Person 1 2 3 4 5 Age (X) 23 19 40 22 35 • Sample mean = (23 + 19 + 40 + 22 + 36)/5 = 140/5 = 28 • And, (23 – 28 + 19 – 28 + 40 – 28 + 22 – 28 + 36 –28) = 140 – 140 = 0. • Verify the results 2 and 3 as well. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Some Concepts • Variable: Any measurable characteristic of a set of data is called a variable. • Discrete Variable: A variable is said to be discrete if it can assume only a finite or countably infinite number of values. • Continuous Variable: A variable is said to be continuous if it can assume any values whatsoever between certain limits. Examples of continuous variables include those representing length, weight, and time. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • Random Experiment: A random experiment is a process leading to at least two possible outcomes with uncertainty as to which will occur. Examples: Tossing a coin, throwing a pair of dice, drawing a card from a pack of cards are all experiments. • Sample Space: The set of all possible outcomes of an experiment is called the sample space (or population). Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • Exercise: Define Sample Space for a) tossing two fair coins, b) tossing three fair coins and c) throwing a pair of dice. • Sample Point: Each member, or outcome of the sample space (or population) is called a sample point. • Event: An event is a subset of the set of possible outcomes of an experiment. Example: Let event A be the occurrence of 'one head and one tail' in the experiment of tossing two fair coins. You can see that only outcomes { HT,TH} belong to event A. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • Mutually exclusive events: Events are said to be mutually exclusive if the occurrence of one event prevents the occurrence of another event at the same time. • Equally likely events: Two events are said to be equally likely if we are confident that one event is as likely to occur as the other event. Example: In a single toss of a coin a head is as likely to appear as a tail. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • Collectively exhaustive: Events are said to be collectively exhaustive if they exhaust all possible outcomes of an experiment. Example: In our two coin tossing experiment {HH,HT,TH,TT} are the possible outcomes, they are (collectively) exhaustive events. • Random Variable (r.v.): A variable that stands for the outcome of a random experiment is called a random variable. A random variable satisfies four properties: – 1) it takes a single, specific value; 2) we do not know in advance what value it happens to take; 3) we do, however know all of the possible values it may take; and 4) we know the probability that it will take any one of those possible values. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • Examples of r.v: The result of rolling of a die, tomorrow's stock price, tomorrow's exchange rate, GNP, money supply, wages, etc. • Discrete r.v.: A random variable that takes a finite number of values is called a discrete random variable. • Continuous r.v.: A random variable that can take any value within a range of values is called a continuous random variable. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • The probability of an event (classical approach): If an experiment can result in n mutually exclusive and equally likely outcomes and m of these outcomes are favourable to event A, then the probability that A occurs ( denoted as P(A)) is the ratio m/n. m the number of outcomes in the event P A n the total number of possible outcomes Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • What happens if the outcomes of an experiment are not finite or not equally likely? • The probability of an event (relative frequency or empirical approach): The proportion of time that an event takes place is called its relative frequency, and the relative frequency with which it takes place “in the long run” is called its probability. If in n trials, m of them are favourable to event A , then P(A) = m/n, provided the number of trials are sufficiently large (technically, infinite). m P A lim . n n Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Probability and Statistics: Basics • There is yet another definition of probability, called as the subjective probability, which is the foundation of Bayesian Econometrics. Under the subjective or “degrees of belief”definition of probability, you can ask questions such as: • What is the probability that Iraq will have a democratic government? • What is the probability that terrorists will attack the United States in November 2004 (the presidential election time)? • What is the probability that there will be a stock market boom in 2005? Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability 1. A probability cannot be negative or larger than 1. That is, 0 P A 1 for any event A. 2. If the probability of the occurrence of event A is P(A), the probability that A will not occur is 1-P(A). 3. Special rule of addition: If A and B are mutually exclusive events, the probability that one of them will occur is P(A or B) = P( A) +P( B). Similarly, if A1,A2 ,...,An are mutually exclusive events, the probability that one of them will occur is P(A1 or A2 or ... or An ) = P( A1 A2 ... An ) P( A1 ) +P( A2 ) ... P( An ). Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability 4. If A1,A2 ,...,An are mutually exclusive and collectively exhaustive set of events, the sum of the probabilities of their individual occurences is 1. That is, P( A1 A2 ... An ) P( A1 ) P( A2 ) ... P( An ) 1. Example:The probability of any of the six numbers on a die is 1/6 since there are six equally likely outcomes and each one of them has an equal chance of turning up. Since the numbers {1,2,3,4,5,6} form an exhaustive set of events P(1+2+3+4+5+6) = P(1)+P(2)+P(3)+P(4)+P(5)+P(6) = 1. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability • Independent Events: Two or more events are said to be independent if the occurrence or non-occurrence of one does in no way affect the occurrence of any of the others. Note: Mutually exclusive events are necessarily independent. However the converse is not necessarily the case. • 5. (Special rule of multiplication) If A and B are independent events, the probability that both of them will occur simultaneously is P(AB) = P(A and B) = P(A).P(B). Since P(A and B) means the probability of events A and B occurring simultaneously or jointly, it is called a joint probability. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability Example: Suppose we flip two identical coins simultaneously. What is the probability of obtaining a head on the first coin (call event A) and a head on the second coin (call event B)? Notice that probability of obtaining a head on the first coin is independent of the probability of obtaining a head on the second coin. Hence, P(AB) = P(A).P(B) = (½).(½) = ¼. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability • 3b (modification to rule 3): If events A and B are not mutually exclusive, then P(A+B) = P(A) + P(B) – P(AB). • Example: A card is drawn from a well shuffled pack of playing cards. What is the probability that it will either a spade or a queen? • Notice that spade and queen are not mutually exclusive events – one of the 4 queens is spade! So, P(a spade or a queen) = P(spade)+P(queen) – P(spade and queen) = 13/52+4/52 – 1/52 = 16/52 = 4/13. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability • Conditional Probability: The probability that event B will take place provided that event A has taken place (is taking place or will with certainty take place) is called the conditional probability B relative to A. Symbolically, it is written as P(B|A) to be read “the probability of B, given A.” • If A and B are mutually exclusive events, then P(B|A) = 0 and P(A|B) = 0. • 6. (General rule of multiplication) If A and B are any two events, then the probability of their occurring simultaneously is P(A and B) = P(A).P(B|A) = P(B).P(A|B). Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Rules of Probability • This implies that P(B|A) = P(AB)/P(A) and P(A|B) = P(AB)/P(B). • Example: In a Principles of Economics class there are 500 students of which 300 students are males and 200 are females. Of these, 100 males and 60 females plan to major in economics. A student is selected at random from this class and it is found that this student plans to be an economics major. What is the probability that the student is a male? • Define A be the event that the student is a male and B be the event that the student is an economics major. Thus, we want to find out P(A|B). • P(A|B) = P(AB)/P(B) = (100/500)/(160/500) = 0.625. (conditional) • What is P(A)? (300/500) = 0.6 (unconditional) Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Exercise 1. The Experiment: Flipping a fair coin three times in a row. • 1a. Let A be the event of getting exactly two heads. What is P(A)? • 1b. Let B be the event of getting a tail on the first flip. What is P(B)? • 1c. Let C be the event of getting no tails. What is P(C)? 2. For the three-flip activity described above, are the two events in each of the following pairs mutually exclusive? • 2a. A and C. • 2b. B and C. • 2c. A and B. • 2d. Two tails, two heads. • 2e. Head on first flip, two tails. • 2f. Tail on first flip, tail on third flip. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Answers Outcomes: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}; denote these outcomes as E1, E2, E3, E4, E5, E6, E7 and E8, respectively. Notice that the probability of each event is 1/8. 1a. Let A be the event of getting exactly two heads. What is P(A)? • P(A) = P(E2)+P(E3)+P(E4) = 1/8 + 1/8 + 1/8 = 3/8. 1b. Let B be the event of getting a tail on the first flip. What is P(B)? • P(B) = P(E4) + P(E6) + P(E7) + P(E8) = 1/8+1/8+1/8+1/8 = ½. 1c. Let C be the event of getting no tails. What is P(C)? • P(C) = P(E1) = 1/8. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Answers (contd.) Outcomes: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}; denote these outcomes as E1, E2, E3, E4, E5, E6, E7 and E8, respectively. Let A be the event of getting exactly two heads: A = {E2, E3, E4} Let B be the event of getting a tail on the first flip: B = {E4, E6, E7, E8) Let C be the event of getting no tails: C = {E1}. Let D be the event of two tails: D = {E5, E6, E7}. Let E be the event of getting head on first flip: E = {E1, E2, E3, E5}. Let F be the event of getting tail on third flip: F = {E2, E5, E6, E8}. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Answers (contd.) A = {E2, E3, E4} B = {E4, E6, E7, E8) C = {E1} D = {E5, E6, E7}. E = {E1, E2, E3, E5} F = {E2, E5, E6, E8} • 2a. A and C – mutually exclusive. • 2b. B and C - mutually exclusive. • 2c. A and B – not mutually exclusive because E4 is common. • 2d. Two tails, two heads: A and D – mutually exclusive. • 2e. Head on first flip, two tails: E and D – not mutually exclusive because E5 is common.. • 2f. Tail on first flip, tail on third flip: B and F – not mutually exclusive because E6 and E8 are common. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Exercise (contd.) A = {E2, E3, E4}, B = {E4, E6, E7, E8) and C = {E1}. 1. What is P(A and B) Answer: P(E4) = 1/8. 2. What is P(A and C) Answer: 3. What is P(B and C) Answer: 4. What is P(A or B)? Answer: P(A) + P(B) – P(AB) = 3/8 + 4/8 – 1/8 = 6/8 = ¾. 5. What is P(A|B)? Answer: P(AB)/P(B) = (1/8)/(4/8) = ¼. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics Exercise (contd.) Let B be the event of getting a tail in the first flip. Let G be the event of getting a tail in the second flip. B = {E4, E6, E7, E8) G = {E3, E5, E7, E8} Question: Are B and G independent? Recall the definition of independent: If A and B are independent events, the probability that both of them will occur simultaneously is P(AB) = P(A and B) = P(A).P(B). That is, A and B are said to be independent, P(A|B) = P(A). Answer: P(B) = 4/8 = ½; P(G) = 4/8 = ½; P(BG) = 2/8; P(G|B) = P(GB)/P(B) = (2/8)/(4/8) = ½. Sisira Sarma 18.317: Introduction to Quantitative Methods in Economics