[This Version: 09-12-2020]1.(Stock & Watson 2019, Exercise 2.1) Let Y denote the number of “heads” that occur when two (fair) coins are tossed.(a)Derive the probability distribution of Y.(b)Derive the cumulative probability distribution of Y.(c)Derive the mean and variance of Y.2.(Stock & Watson 2019, Exercise 2.5) In September, Seattle’s daily high temperature has a mean of 70 degrees Fahrenheit and a standard deviation of 7 degrees Fahrenheit. What are the mean, standard deviation, and variance in degrees Celsius?3.(Stock & Watson 2019, Exercise 2.9) X and Y are discrete random variables with the following joint distribution:Y =14 Y =22 Y =30 Y =40 Y =65X=1 0.02 0 05 0.10 0 03 0.01X=5 0.17 0 15 0.05 0 02 0.01X=8 0.02 0 03 0.15 0 10 0.09This is, Pr(X = 1,Y = 14) = 0.02, and so forth.(a)Calculate the probability distribution, mean, and variance of Y .(b)Calculate the probability distribution, mean, and variance of Y given X = 8.(c)Calculate the covariance and correlation between X and Y .4.(Stock & Watson 2019, Exercise 2.10) Compute the following probabilities:(a)If Y is distributed N(1,4), find Pr(Y ≤ 3). (b) If Y is distributed N(3,9), find Pr(Y > 0).(c)If Y is distributed N(50,25), find Pr(40 ≤ Y ≤ 52).(d)If Y is distributed N(5,2), find Pr(6 ≤ Y ≤ 8).

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