STAT 210: PROBLEM SET 5 VERSION A (CAN BE DISCUSSED IN OFFICE HOURS) DUE: 10-20 1. The probability distribution of random variable X is given below: x 2 P(x) 0.9 What is 2X (rounded to four decimal places)? 4 0.1 2. The probability distribution of random variable X is given below: x 2 4 P(x) 0.9 0.1 A second random variable is defined as follows: Y = 4X - 5. What is 2Y (rounded to four decimal places)? 3. The probability distribution of random variable X is given below: x 1 2 P(x) 0.4 0.6 A second random variable is defined as follows: Y = 3X - 2. What is Y (rounded to four decimal places)? 4. The probability distribution of random variable X is given below: x 1 2 P(x) 0.4 0.6 A second random variable is defined as follows: Z = 3X2 - 1. What is Z (rounded to four decimal places)? 5. The probability distribution of random variable X is given below: x 1 2 P(x) 0.2 0.8 A second random variable is defined as follows: Z = 10 - X2. What is 2Z (rounded to four decimal places)? 6. Natalie converts 65% of her free throws. (Her free throws are independent.) She goes to the line to shoot a \"two-shot foul.\" (Whether or not the first shot is made, she is allowed a second shot. Each successful shot is worth one point.) The random variable X is the number of points resulting from Natalie's trip to the line for a \"two-shot foul.\" What is E(X)? (Round your answer to four decimal places.) 7. Jen converts 75% of her free throws. (Her free throws are independent.) She goes to the line to shoot a \"one and one.\" (If the first shot is successful, she is allowed a second shot, but no second shot is taken if the first shot is missed. Each successful shot is worth one point.) The random variable Y is the number of points resulting from Jen's trip to the line for a \"one and one.\" What is E(Y)? (Round your answer to four decimal places.) 8. The Cubs are to play a series of 5 games with the Cardinals. For any one game it is estimated that the probability of a Cubs' win is 0.35. The outcomes of the 5 games are independent. Let X be the number of games that the Cubs win in the series. What is X? (Round your final answer to four decimal places if necessary.) 9. The Cubs are to play a series of 3 games with the Cardinals. For any one game it is estimated that the probability of a Cubs' win is 0.45. The outcomes of the 3 games are independent. Let X be the number of games that the Cubs win in the series. What is the probability that the Cubs win a majority of the games in the series? (Round your final answer to four decimal places if necessary. 10. Following a touchdown, a college football coach has the option to elect to attempt a twopoint conversion; that is, two additional points are scored in the attempt is successful and none, if it is unsuccessful. The coach believes that the probability is 0.35 that his team will be successful in any attempt and that the outcomes of different attempts are independent of each other. In a particular game the team scores 4 touchdowns and two-point conversion attempts were made each time. a. What is the probability that at least 2 of these attempts will be successful? b. Find the mean and standard deviation of the total number of points resulting from these 4 attempts. c. What is the required success rate for completing one-point conversions so that the same mean number of points are obtained? \f