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Section 2

# Section 2 - Section2 Econ I40 GSI Edson Severnini...

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Section2- Econ I40 GSI: Edson Severnini DERryATION OF LEAST SQUARES Minimize j ,n - f,), fhs least-squales solutions for thg slope and intercept are w h e r e ? , : o + b X , hlhen the sample means of X and Y are 0, a:T - bX : o b - N >XiYi - >xi>Yj N >X1 - (2x,)' >&vrlN - (>xrl\$(>rrlN) >x1tN - (>&/N)' >xiYilN - xY ZVTN -Z >xiYitN ZxiYi txW: >x1 N>XiYi - >xt>Y, - N >x1 _ (>x), o :+ - b2x, : y - bx N " N b - least-squares line Y = a + b X

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Now ,ruriables that are exPressed least-squares sloPe estimate is as deviations from their respective means have meafls. zero. Thus, the w h e r e x i : X i - X l i : Y i - Y , Zxiyi 0: >E GRADE-POINT.AVERAG E EXAM PLE (CALCULATIONS) X : 1 3 . 5 7 = 3 . 0 (1) X i = X i - X (2) Y r = Y i - Y (3) Xili (4) xf 7.5 '1.5 1 . 5 -4.5 - 1 . 5 4.5 - t.5 - 1 . 5 ) x , : o 56.25 2.25 2.25 20.25 2.25 20.25 56.25 2.25 1.0 .0 - 1.0 .0 . 5 - . 5 - . 5 2 Y i = o . 1 2 0 a : Y - O X 7.5 .0 .75 4.5 .0 2.25 3.75 .75 2x;y, = 19.59 : 1.375 ?: l.ezS + .lZ Zxl: 162.0, ^ -2x, Y, - u - D x ? - i a 0.) tr3 OJ (! .=) c., (o u l Y = 1 . 3 7 5 + . 1 2 X 6 1 2 1 8 2 4 Family income (in thousands of dollars) (a) Original Regression Line RANDOM VARIABLES Definition: A random variable is equalto 1. Types of random variable: 'L0 .75 .50 .25 (b) Transformed Regression x = X - 1 3 . 5 Y = Y - 3 ' o a variable that takes alternative values, each with a probability less than or o Continuous: takes any value on the real line o Discrete: takes only specific numbers on the real line = o) = .; (! = = .g i . 0 .75 .50 .25
Probability distribution: process that generates the values of a random variable. o Probability densitv function (p.d.f.): lists all possible outcomes of a random ated probabilities lP(X: 0)] o Cumulative distribution function k.d.f.): cumulates the total probability until a certain value of the random variable lP(X S z)] Also, probability distributions are often described in terms of their means and variances, which in their turn are defined in terms of the expectation operator .8. o Mean. or expected value, of X : Fx : E(X) : ptXt * pzXz+ ... + pxXN : f pnX, i : 1 r Variance of X: ofu : Var(X) : nlV - tr; 'l : f on6o - p,il2 L ' " ' J i - = r ' - the positive square root of the variance is called standard deviation (a76) Resultl E(aX + b): aE(X) + b where X is a random variable and a and b are constants.

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Section 2 - Section2 Econ I40 GSI Edson Severnini...

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