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6 - Lecture 2

# 6 - Lecture 2 - Introduction to Econometrics Econ 322...

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Introduction to Econometrics Econ 322; Summer 2011 Ruby HENRY [email protected] June 6, 2011 Ruby HENRY () Introduction to Econometrics June 6, 2011 1 / 33

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Moments of a Random Variable A random variable X (assume from now on that X is a discrete RV) has a probability distribution function F(x) where < 2- > Prob ( X = x ) = F ( x ) Ruby HENRY () Introduction to Econometrics June 6, 2011 2 / 33
Moments of a Random Variable A random variable X (assume from now on that X is a discrete RV) has a probability distribution function F(x) where < 2- > Prob ( X = x ) = F ( x ) Ruby HENRY () Introduction to Econometrics June 6, 2011 2 / 33

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Moments of a Random Variable A random variable X (assume from now on that X is a discrete RV) has a probability distribution function F(x) where < 2- > Prob ( X = x ) = F ( x ) Let Z be the set of all possible values of X . Then the probability distribution function, F, is de°ned < 4- > for all x 2 Z , F ( x ) ° 0 < 5- > x 2 Z F ( x ) = 1 Ruby HENRY () Introduction to Econometrics June 6, 2011 2 / 33
Example the expected value of the value showing when you toss a die is Ruby HENRY () Introduction to Econometrics June 6, 2011 3 / 33

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Example the expected value of the value showing when you toss a die is E ( X ) = 1 ± 1 / 6 + 2 ± 1 / 6 + 3 ± 1 / 6 + 4 ± 1 / 6 + 5 ± 1 / 6 + 6 ± 1 / 6 = 3 . 5 Ruby HENRY () Introduction to Econometrics June 6, 2011 3 / 33
Example the expected value of the value showing when you toss a die is E ( X ) = 1 ± 1 / 6 + 2 ± 1 / 6 + 3 ± 1 / 6 + 4 ± 1 / 6 + 5 ± 1 / 6 + 6 ± 1 / 6 = 3 . 5 the variance is var ( X ) = E (( X ² E ( X )) 2 ) = ( 1 ² 3 . 5 ) 2 ± 1 / 6 + . . . + ( 6 ² 3 . 5 ) 2 ± 1 / 6 = 2 . 91 Ruby HENRY () Introduction to Econometrics June 6, 2011 3 / 33

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Properties of Mean and Variance F Suppose we have a new random variable that is a linear function of the random variable X Ruby HENRY () Introduction to Econometrics June 6, 2011 4 / 33
Properties of Mean and Variance F Suppose we have a new random variable that is a linear function of the random variable X e.g. Y = a + bX Ruby HENRY () Introduction to Econometrics June 6, 2011 4 / 33

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Properties of Mean and Variance F Suppose we have a new random variable that is a linear function of the random variable X e.g. Y = a + bX what are the moments of Y? Ruby HENRY () Introduction to Econometrics June 6, 2011 4 / 33
Properties of Mean and Variance F Suppose we have a new random variable that is a linear function of the random variable X e.g. Y = a + bX what are the moments of Y? (long way) we can compute the probability density function of Y Ruby HENRY () Introduction to Econometrics June 6, 2011 4 / 33

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Properties of Mean and Variance F Suppose we have a new random variable that is a linear function of the random variable X e.g. Y = a + bX what are the moments of Y? (long way) we can compute the probability density function of Y y=a+bx a+b a+2b a+3b a+4b a+5b a+6b F(y) 1/6 1/6 1/6 1/6 1/6 1/6 Ruby HENRY () Introduction to Econometrics June 6, 2011 4 / 33
Properties of Mean and Variance(Cont) mean is E ( Y ) = y 2Y yF ( y ) which is equal to E ( Y ) = ( a + b ) ± 1 / 6 + ( a + 2 b ) ± 1 / 6 + ( a + 3 b ) ± 1 / 6 + ( a + 4 b ) ± 1 / 6 + ( a + 5 b ) ± 1 / 6 + ( a + 6 b ) ± 1 / 6 = a + ( 1 ± 1 / 6 + 2 ± 1 / 6 + 3 ± 1 / 6 + 4 ± 1 / 6 + 5 ± 1 / 6 + 6 ± 1 / 6 ) b = a + 3 . 5 b Ruby HENRY () Introduction to Econometrics June 6, 2011 5 / 33

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Properties of Mean and Variance(Cont) mean is
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6 - Lecture 2 - Introduction to Econometrics Econ 322...

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