ECON2280_2018_sem1_Tutorial 1 PS.pdf - ECON2280 Introductory Econometrics Tutorial 1 Tutorial 1 Problem set Question 1(Review of Statistics Here is the

# ECON2280_2018_sem1_Tutorial 1 PS.pdf - ECON2280...

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ECON2280 Introductory Econometrics Tutorial 1 Tutorial 1 Problem set Question 1 (Review of Statistics) Here is the table describing the joint distribution of the random variable X and Y. Pr ( X = x; Y = y ) Y = 4 Y = 9 X = 1 0.24 0.16 X = 2 0.12 0.08 X = 4 0.24 0.16 1. Calculate the marginal probability of X and the marginal probability of Y. 2. Find E ( X ) and E ( Y ) : 3. Find V ar ( X ) and V ar ( Y ) : 4. De°ne a new random variable Z = XY: Find out the distribution of Z . 5. Find E ( Z ) and V ar ( Z ) : 6. Based on the joint probabilities and the marginal probabilities, are X and Y independent of each other? 7. Find Cov ( X; Y ) using (a) the formula Cov ( X; Y ) = E [( X ° E ( X )) ( Y ° E ( Y ))] : 1 1. P(X=1) = 0.24+0.16 = 0.4 P(X=2) = 0.12+0.08 = 0.2 P(X=4) = 0.24+0.16 = 0.4 P(Y=4) = 0.24+0.12+0.24=0.6 P(Y=9) = 0.16+0.08+0.16=0.4 2. E(X)= 1x0.4 +2x0.2 +4x0.4 =2.4 E(Y)= 4x0.6 + 9x0.4= 6 3. Var(X)= 0.4x(1-2.4)^(2) + 0.2x(2-2.4)^(2) + 0.4x(4-2.4)^(2)= 1.84 Var(Y) = 0.6x(4-6)^(2) + 0.4x(9-6)^(2) = 6

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(b) the formula Cov ( X; Y ) = E ( XY ) ° E ( X ) E ( Y ) : 8. Find Corr ( X; Y ) 9. Find E ( X j Y = 4) ; E ( Y j X = 4) ; V ar ( X j Y = 4) ; V ar ( Y j X = 4) : 10. Find the conditional probabilities of Y given Z > 8 ; i:e:; Pr ( Y = 4 j Z > 8) and Pr ( Y = 9 j Z > 8) , where Z = XY: Question 2 1. Using the fact that

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