1. A small data analysis class has five computer science majors among the twenty-two total students. Three people are chosen at random to form a group. Let X denote the number of computer scientists in this group. (a) Find E(X). (b) Find Var(X). (c) Find the standard deviation of X. 2. The ages (to the nearest year) of students enrolled in a specific data analysis class have a cumulative probability distribution given by 17 18 19 20 21 22 23 F(x) 0 0.05 0.28 0.60 0.83 0.92 1.00 (a) Find the mean of X. (b) Find the standard deviation of X. 3. Suppose that discrete random variables X and Y have the following incomplete joint distri- bution table. Y 0 X Px (X) -1 0.18 0.6 Py ( y) 0.3 (a) Complete the table above. Are X and Y independent? Briefly explain how you deter- mined this. (b) Find E(X + Y). (c) Find Var(X + Y). 4. The New York Rangers win 63.4% of their games. Suppose they play seven games against the New York Islanders in a season. What is the probability that the Rangers end up winning the majority of their games against the Islanders this season? 5. At the beginning of 2022, there were 19 women in the NY Senate, versus 44 men. Suppose that a five-member committee is selected at random. Calculate the probability that the committee has a majority of women. 6. Suppose that the probability that someone in NYC has been exposed to monkeypox is 10%. (a) Find the probability that at most 3 out of 20 randomly chosen people in NYC have been exposed to monkeypox. (b) Find the probability that one must test exactly 20 people in order to find 3 people that have been exposed to monkeypox. 7. The east coast suffered seven major hurricanes in 2021. Assume that this is the new average, and that the occurrence of hurricanes follows a Poisson distribution. Find the probability that in the next two years, the east coast will be hit by a number of hurricanes between 15 and 18 (inclusive)