ps1-2

Eco 240: Econometrics Professor Mariyana Zapryanova Problem Set 1 Due: September 19 Directions: You must show your work on all questions to receive full credit. Even though you may be able to take short cuts in your head, please write down all steps! Problem sets must be written up neatly, either by hand or typed. Please clearly designate where you are showing work and where your final answer is located (e.g., circle or highlight your final answer) or you may not receive credit. If your answer cannot be read or found, you may not receive credit. 1. Consider the situation of rolling two dice and let M denote a random variable that is the sum of the number of dots on the two dice. (M is a number between 1 and 12.) Assume that the two dice are independent (standard assumption). a) In the table below, list all the possible outcomes for the random variable M together with its probability distribution and cumulative probability distribution. Explain how you calculated the numbers. Outcome 2 3 4 5 6 7 8 9 10 11 12 Probability distribution 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Cumulative probability distribution 1/36 3/36 6/36 10/36 15/36 21/36 26/36 30/36 33/36 35/36 36/36 b) Use your table to calculate the probability of the following outcomes, explaining your answer. Pr(M = 2 or M = 10) Pr(M < 8) Pr(M=2) = 1/36 Pr(M=10) = 3/36 Pr(M=2 or M=10) = 1/36 + 3/36 = 1/9 Pr(M < 8) = 1/36 + 2/36 + 3/36 + 4/36 + 5/36 + 6/36 = 21/36 c) Calculate the expected value and the standard deviation of M. Show your work fully! E(M) = (2*1/36) + (3*2/36) + (4*3/36) + (5*4/36) + (6*5/36) + (7*6/36) + (8*5/36) + (9*4/36) + (10*3/36) + (11*2/36) + (12*1/36) = Standard deviation

M M - u (M – u)2 P(M) (M – u)2 * p(M) 2 -5 25 1/36 3 -4 16 2/36 4 -3 9 3/36 5 -2 4 4/36 6 -1 1 5/36 7 0 0 6/36 8 1 1 5/36 9 2 4 4/36 10 3 9 3/36 11 4 16 2/36 12 5 25 1/36 Mean = 7 0 110 1 2. Assume that the height of male students at Amherst is normally distributed with a mean of 70 inches and a standard deviation of 3.5 inches. If you had a list of telephone numbers for male students for the purpose of conducting a survey, what would be the probability of randomly calling one of those students whose height is the following values (6 pts): a) shorter than 5'7"? Z = 67 – 70 / 3.5 = b) taller than 6'0"? c) between 5'3" and 6'5"? d) taller than Shaquille O'Neal, who is 7'1" tall? Note: you can use the CDF table 1 in the book to look up the probabilities from the standard normal function. Show your work/ explain. (NOTE: you can also check your work in Stata using the command “display normal(Z)” where Z is the Z score and Stata will display the CDF value.) 3. The table below gives the joint probability distribution between employment status (Y) and whether a person suffers from alcohol use disorder (X). Joint Distribution of Employment Status and Alcohol Abuse Unemployed (Y = 0) Employed (Y = 1) Total No Alcohol Use Disorder (X = 0) 0.089 0.811 0.900 Presence of Alcohol Use Disorder (X = 1) 0.013 0.087 0.100 Total 0.102 0.898 1.000 a) Let Y denote a variable, which takes on a value of 1 if the person is employed, and 0 if unemployed. Let X denote a variable which takes on a value of 1 if the person suffers from alcohol use disorder and 0 if not. Compute E(Y). b) Calculate E(Y|X = 1) and E(Y|X=0).