12. The two largest cable providers are Comcast Cable Communications, with 21.5 million subscribers, and Time Warner Cable, with 11.0 million subscribers (The New York Times Almanac, 2007). Suppose that the management of Time Warner Cable subjectively assesses a probability distribution for the number of new subscribers next year in the state of New York as follows. x f(x) 100,000 .10 200,000 20 300,000 25 400,000 30 500,000 .10 600,000 .05 a. Is this probability distribution valid? Explain. b. What is the probability Time Warner will obtain more than 400,000 new subscribers? c. What is the probability Time Warner will obtain fewer than 200,000 new subscribers?17. The number of students taking the SAT has risen to an all-time high of more than 1.5 million (College Board, August 26, 2008). Students are allowed to repeat the test in hopes of improving the score that is sent to college and university admission offices. The number of times the SAT was taken and the number of students are as follows. E Number of Number of Times Students 721,769 601,325 UAWNE 166,736 22,299 6730 a. Let x be a random variable indicating the number of times a student takes the SAT. Show the probability distribution for this random variable. b. What is the probability that a student takes the SAT more than one time? c. What is the probability that a student takes the SAT three or more times? d. What is the expected value of the number of times the SAT is taken? What is your interpretation of the expected value? e. What is the variance and standard deviation for the number of times the SAT is taken?25. Given below is a bivariate distribution for the random variables x and y. f(x, y) x 50 80 30 40 60 a. Compute the expected value and the variance for x and y. b. Develop a probability distribution for x + y. c. Using the result of part (b), compute E(x + y) and Var (x + y). d. Compute the covariance and correlation for x and y. Are x and y positively related, negatively related, or unrelated? e. Is the variance of the sum of x and y bigger, smaller, or the same as the sum of the in- dividual variances? Why?13. A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1, 2, or 3. Let x be a random variable indicating the number ofsessions required to gain the patient's trust. The following probability function has been proposed. f (x ) = X 6 for x = 1, 2, or 3 a. Is this probability function valid? Explain. b. What is the probability that it takes exactly 2 sessions to gain the patient's trust? c. What is the probability that it takes at least 2 sessions to gain the patient's trust
