stats_ps5_winter24

Stats Problem Set #5 1. The probability distribution of random variable X is given below: x 2 20 P ( x ) 0.4 0.6 What is ࠵? ࠵? (rounded to four decimal places)? 2. The probability distribution of random variable X is given below: x 2 10 P ( x ) 0.8 0.2 A second random variable is defined as follows: Y = 4 X – 5. What is ࠵? ࠵? (rounded to four decimal places)? 3. The probability distribution of random variable X is given below: x 1 5 P ( x ) 0.9 0.1 A second random variable is defined as follows: Y = 3 X – 2. What is ࠵? ࠵? 2 (rounded to four decimal places)? E(X) = I (0 8) + 20 (0 . 6) = 12 . 8 Ox = (2 - 12 . 8)(0 8) A(20 - 12 6) (6 . 6) 2 A (2 . 0 8) E(b) = 3 (0 . 8) + 35 (0 2) = 9 . & O = (3 - 9 . )(0 8) + (38 - 9 . 4) (0 . 2) O = 3 . Z & 181c . 2 : T ECO) = H (0 . 9) + 13 (6 . 1) = 2 . 2 o = (n - 2 . 25 20 9) + (13-2 2) (01) = T

4. 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 = 4 X 2 – 1. What is ࠵? ࠵? 2 (rounded to four decimal places)? 5. A shipment of 16 parts contains 2 defective parts. Three parts are chosen at random from the shipment and checked. What is the probability that exactly one defective part is found? (Remember to express your answer as a proportion. Round your final answer to four decimal places if necessary.) 6. Natalie converts 56% 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.)

9. (A touchdown is worth 6 points. An extra point is worth 1 point. A two-point conversion is worth 2 points. After a team scores a touchdown, they can either attempt an extra point or attempt a two-point conversion. The probability that an extra point attempt is converted is 0.97. The probability that a two-point conversion attempt is converted is 0.46. All extra point attempts and two-point conversion attempts are independent.) The Bears are down by 14 points late in the fourth quarter to the Packers. Let’s assume that the only way the Bears can win is to score 14 points (via two touchdowns and two extra points or two touchdowns and one two-point conversion) and win in overtime, or to score 15 points (via two touchdowns, an extra point, and a two-point conversion) and win in regulation. Let’s stipulate that the Bears do score the required two touchdowns in the final minutes and do not give up any more points to the Packers, and let’s assume that each team has an equal chance of winning in overtime should the game go that far. Given these stipulations and assumptions, what is the probability that the Bears will win if they pursue the following strategy: go for two after the first touchdown and then only go for two after the second touchdown if the first two-point conversion attempt fails. (Round your final answer to four decimal places if necessary.) 10. (A touchdown is worth 6 points. An extra point is worth 1 point. A two-point conversion is worth 2 points. After a team scores a touchdown, they can either attempt an extra point or attempt a two-point conversion. The probability that an extra point attempt is converted is 0.96. The probability that a two-point conversion attempt is converted is 0.44. All extra point attempts and two-point conversion attempts are independent.) The Bears are down by 14 points late in the fourth quarter to the Packers. Let’s assume that the only way the Bears can win is to score 14 points (via two touchdowns and two extra points or two touchdowns and one two-point conversion) and win in overtime, or to score 15 points (via two touchdowns, an extra point, and a two-point conversion) and win in regulation. Let’s stipulate that the Bears do score the required two touchdowns in the final minutes and do not give up any more points to the Packers, and let’s assume that each team has an equal chance of winning in overtim e should the game go that far. Given these stipulations and assumptions, what is the probability that the Bears will win if they pursue the following strategy: go for one after the first touchdown and then only go for two after the second touchdown if the first extra-point attempt fails. (Round your final answer to four decimal places if necessary.)