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 where?,:o+bX, 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 least-squares line Y=a+bX
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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 where xi:Xi-X li:Yi-Y , Zxiyi 0: >E GRADE-POINT.AVERAG E EXAM PLE (CALCULATIONS) X:13.5 7=3.0 (1) Xi=Xi-X (2) Yr=Yi-Y (3) Xili (4) xf 7.5 '1.5 1.5 -4.5 - 4.5 t.5 )x,:o 56.25 2.25 20.25 1.0 .0 .5 -.5 2Yi =o .120 a:Y OX .75 3.75 2x;y, = 19.59 : 1.375 ?: l.ezS + .lZ Zxl: 162.0, ^ -2x, Y, - u- Dx?- i a 0.) tr3 OJ (! .=) c., (o ul Y=1.375+.12X 6121824 Family income (in thousands of dollars) (a) Original Regression Line RANDOM VARIABLES Definition: A random variable is equal to 1. Types of random variable: 'L0 .50 .25 (b) Transformed Regression x = X- 13.5 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
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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. Result2 El(axlzl: a2E(X2) Result 3 Yar (aX + b\ a2 Yar (X) o Covariance'. oyx Cou(Y,X) El(Y Hy) (x px)l: & p'v) (Xi px) (E(x))' Recall: Cou(Y,X) E(YX) E(Y)E(X) =+ Var(X) It can be described by: variable with their associ- \-\--.
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This note was uploaded on 08/06/2009 for the course ECON 140 taught by Professor Duncan during the Summer '08 term at University of California, Berkeley.

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

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