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samplequestions1 (1)

# samplequestions1 (1) - Exercises to Help Study for Midterm...

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Exercises to Help Study for Midterm 1 Part I: Probability Question 1 Table 1: Joint Distribution of Weather Conditions and Commuting Times Rain (X=0) No Rain (X=1) Total Long Commute (Y=0) 0.15 0.07 0.22 Short Commute (Y=1) 0.15 0.63 0.78 Total 0.30 0.70 1.00 1) Use the probability distribution given in Table 1 to compute (a) E(Y) and E(X); (b) σ 2 X and σ 2 Y ; and (c) σ XY and corr(X,Y). 2) Using the random variables X and Y from the same table, consider two new random variables W = 3 + 6X and V=20 – 7Y. Compute (a) E(W) and E(V); (b) σ 2 W and σ 2 V ; and (c) σ WV and corr(W,V). Question 2 Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that ** ( , ) ( , ) corr X Y corr X Y , where * X a bX  and * Y c dY , and where a , b , c , and d are arbitrary non-zero constants. Question 3 The table below shows the joint probability distribution employment status (employed and unemployed) and education (no college degree and college degree) in an economy. a) Fill in the missing entries in the table. b) Compute the economy’s unemployment rate (Pr( X = 1)) c) Compute E ( Y ) and Var (Y ). Question 4 Compute the following probabilities:

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a) If Y is distributed N (1,2), find Pr( Y ≤ 3). b) If Y is distributed N (1,9), find Pr( Y > 0). c) If Y is distributed N (5,100), find Pr(0 ≤ Y ≤ 7). d) If Y is distributed N (15,16), find Pr( Y > 12). Question 5 In a population such that the population mean μ = 10 and variance σ 2 = 9. Use the central limit theorem to answer the following questions: a) In a random sample of size n = 100, find Pr( Y ≤ 10.5).
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samplequestions1 (1) - Exercises to Help Study for Midterm...

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