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Lecture 3 Prof Arkonac's slides (Ch 2,3 and a little 4) for Eco 4000

# Lecture 3 Prof Arkonac's slides (Ch 2,3 and a little 4) for Eco 4000

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ECO 4000, Statistical Analysis for Economics and Finance Fall 2010 Lecture 3 Prof: Seyhan Arkonac, PhD

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Last Lecture we talked about expected value, mean and variance. Let’s remember them:
Expected Value, Mean and Variance Let X be a random variable (say # of Heads when you flip a coin twice) E(X) = ∑ X Prob(X)= (0)(.25)+(1)(.50)+(2)(.25)=1 Var(X) = E(X 2 ) [E(X)] 2 E(X 2 ) = (0 2 ) (.25) + (1 2 )(.50) + (2 2 ) (.25) = 1.50 [E(X)] 2 = 1 2 = 1 Var(X) = E(X 2 ) [E(X)] 2 = 1.50 1 = 0.50

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2- 4
2- 5

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How about correlation coefficient? What is correlation coefficient between two random variables? Say the correlation coefficient between (1) class size and (2) student performance
7 The correlation coefficient is defined in terms of the covariance: corr( X , Z ) = cov( , ) var( )var( ) XZ X Z X Z X Z   = r XZ 1 corr( X , Z ) 1 corr( X , Z ) = 1 mean perfect positive linear association corr( X , Z ) = 1 means perfect negative linear association corr( X , Z ) = 0 means no linear association

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What was covariance? Remember?
9 2 random variables: covariance The covariance between X and Z is cov( X , Z ) = E [( X X )( Z Z )] = XZ The covariance is a measure of the linear association between X and Z ; its units are units of X units of Z cov( X , Z ) > 0 means a positive relation between X and Z If X and Z are independently distributed, then cov( X , Z ) = 0 (but not vice versa!!) The covariance of a r.v. with itself is its variance cov( X , X ) = E [( X X )( X X )] = E [( X X ) 2 ] = 2 X

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Now, let’s see again “pictures” of some possible correlations between 2 random variables: Remember: 1 corr( X , Z ) 1 corr( X , Z ) = 1 mean perfect positive linear association corr( X , Z ) = 1 means perfect negative linear association corr( X , Z ) = 0 means no linear association
11 The correlation coefficient measures linear association

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Here was the plan (from last lecture): Review of Statistical Theory: (1) The Probability Framework for Statistical Inference (2) Estimation (3) Testing (4) Confidence Intervals
(1) The Probability Framework for Statistical Inference (a) Population, random variable and distribution (b)Moments of distribution (mean, variance, standard deviation, covariance, correlation (c) Conditional distributions and conditional means (d)Distribution of a sample of data randomly from a population: Y 1 …….Y n ( we did part (a) and part (b) last time, now let’s continue from part (c))

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14 (c) Conditional distributions and conditional means Conditional distributions The distribution of Y , given value(s) of some other random variable, X Ex: the distribution of test scores, given that STR < 20 Conditional expectations and conditional moments conditional mean = mean of conditional distribution = E ( Y | X = x ) ( important concept and notation ) conditional variance = variance of conditional distribution Example : E ( Test scores | STR < 20) = the mean of test scores among districts with small class sizes The difference in means is the difference between the means of two conditional distributions: