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Unformatted text preview: Kata Bognar kbognar@ucla.edu Economics 41 Statistics for Economists UCLA Fall 2010 Handout  Week 1 1 Math Review I Summation notation. (See Rogawski, Single Variable Calculus p 250251.) 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, nonnegative 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 xaxis 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 nonnegative functions. 1 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 ( 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 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....
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 Fall '07
 Guggenberger
 Economics

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