[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 =65
X=1 0.02 0 05 0.10 0 03 0.01
X=5 0.17 0 15 0.05 0 02 0.01
X=8 0.02 0 03 0.15 0 10 0.09
This 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).

Sample Solution