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# ec41handout1 - Kata Bognar [email protected] Economics 41...

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Kata Bognar [email protected] Economics 41 Statistics for Economists UCLA Fall 2010 Handout - Week 1 1 Math Review I Summation notation. (See Rogawski, Single Variable Calculus p 250-251.) We are often interested in the sum of several terms and in such cases a compact notation is very useful. The symbol n j =1 a j for 1 n denotes the sum of the terms a 1 , a 2 , . . . , a n . Or equivalently, n X j =1 a j = a 1 + a 2 + · · · + a n . Example 1: 4 j =1 j = 1 + 2 + 3 + 4 = 10 . Example 2: 4 j =1 2 j = 2 + 4 + 6 + 8 = 20 . Example 3: 4 j =1 j 2 = 1 + 4 + 9 + 16 = 30 . The summation notation refers to sums so the properties of addition remain true. 1. n j =1 Ca j = Ca 1 + Ca 2 + · · · + Ca n = C ( a 1 + a 2 + · · · + a n ) = C n j =1 a j , for any C constant 2. commutativity Example 4: 4 j =1 2 j = 2 4 j =1 j = 2 · 10 = 20 . Example 5: 6 j =1 = 1 + 2 + 3 + 4 + 5 + 6 = (1 + 3 + 5) + (2 + 4 + 6) = 3 j =1 (2 j - 1) + 3 j =1 (2 j ) . We often use the letter X to refer to a quantitative variable (see definition later) and the symbols x 1 , x 2 , . . . , x n refer to different observations of this variable. Then the sum of the observations is denoted by X i x i x 1 + x 2 + x 3 + · · · + x n . This notation will be convenient whenever we think about descriptive measures (see definition later) for the data set such as mean, standard deviation, etc. Example 6: We have data about the weekly salary of a group of people. The salary is a quantitative variable; we denote the salary by X. Consider the following 10 observations on the weekly salaries: 300 300 940 450 400 400 300 300 1050 300 Denote x 1 = 300 , x 2 = 300 , x 3 = 940 , x 4 = 450 , x 5 = 400 , . . . , x 10 = 300 . The sum of the observations is i x i = x 1 + x 2 + x 3 + . . . x 10 = 4740 . Integral. (See Rogawski, Single Variable Calculus Chapter 5,7.) Suppose that f ( x ) : R R is a continuous, non-negative function. Then the integral of f ( x ), Z b a f ( x )d x can be interpreted as the area of the region between the graph of the function and the x-axis over the interval [ a, b ] . We will use the following rules in calculating probabilities, so please review them. Suppose that f ( x ) and g ( x ) are both continuous and non-negative functions. 1

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1. R b a C d x = C ( a - b ) for any C constant 2. R b a ( f ( x ) + g ( x ))d x = R b a f ( x )d x + R b a g ( x )d x 3. R c a f ( x )d x = R b a f ( x )d x + R c b f ( x )d x 4. (Change of variables) R b a f ( u ( x )) u 0 ( x )d x = R u ( b ) u ( a ) f ( u )d u 5. R x n d x = x n +1 n +1 + C if n 6 = - 1 6. R x - 1 d x = ln | x | + C 7. R e x d x = e x + C 8. R a x d x = a x ln a + C Example 1: R 1 0 1d x = 1(1 - 0) = 1 Example 2: R 2 x ( x 2 + 9) 5 d x = 1 6 ( x 2 + 9) 6 + C (i) define u = ( x 2 + 9) (ii) use change of variables formula (iii) use the formula for the exponential functions (iv) substitute back u = ( x 2 + 9) 1.1 Practice Problems. 1. Evaluate the following integrals: (a) R 6 3 x 2 d x (b) R 4 0 x +2 18 d x (c) R 2 0 x 2 x 3 + 1d x 2 Basic Concepts A population is the collection of all items under consideration in the study. A sample is the part of the population about which information is collected. An element (of a population or a sample) is a specific subject about which the information is collected. A variable is a characteristics that varies from one elements to another. Suppose that we are interested in the GPA of the Econ 41 students of Fall 10. Then the population of interest is all students in the class, a sample would be a selected subset
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ec41handout1 - Kata Bognar [email protected]u Economics 41...

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