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HOW People Getl heir News The Brunswick Research Organization surveved 50 randomly,r selected individuals and asked them the pmal'f wag.r they received the daily news. Their choices were via newspaper {N}, television (Ti. radio (R), or lntemet {I}. Construct a categorical frequency.r distribution forthe data and interpret the results. The data in this exercise will be used for Exercise 2 in this secon. [21] Refer to Exercise 12 and calculate V(Y) and sY. Then determine the probability that Y is within 1 standard deviation of its mean value. Reference exercise -12 Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. a. What is the probability that the flight will accommodate all ticketed passengers who show up? b. What is the probability that not all ticketed passengers who show up can be accommodated? c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?A result called Chebyshev's inequality states that for any probability distribution of an rv X and any number k that is at least 1, P( | X - p | k o) s 1/k . In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/12. a. What is the value of the upper bound for k = 2? K + 3? K = 4? K= 5? K = 10? b. Compute p and o for the distribution of Exercise 13. Then evaluate P(IX - p| > ko) for the values of k given in part (a). What does this suggest about the upper bound relative to the corresponding probability? c. Let X have possible values -1, 0, and 1, with probabilities 1/18, 8/9 and 1/8 , respectively. What is P(IX - p/2 30), and how does it compare to the corresponding bound? d. Give a distribution for which P(IX - p|2 50) = .04. Reference exercise -13 A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. P(| X - # | > ko) s 1/k2. Calculate the probability of each of the following events. a. {at most three lines are in use} b. {fewer than three lines are in use} c. {at least three lines are in use} d. {between two and five lines, inclusive, are in use} e. {between two and four lines, inclusive, are not in use} f. {at least four lines are not in use}The Great Lakes Shown are 1uran'ous statistics about the Great Lakes. Using appropriate graphs [your choice} anti summary.r statements, write a report analyzing the data. Length {miles} - M M Breadth (miles) 113 Volume {cubic miles) __ 393 Area {square miles]: 530 3&0 Shoreline (LI.S_, miles]: 1,400 - 1550 Vehicle speed on a particular bridge in China can k modeled as normally distributed ("Fatigue Reliability Assessment for Long-Span Bridges under Combined Dynamic Loads from Winds and Vehicles," J. of Bridge Engr., 2013: 735-747). a. If 5% of all vehicles travel less than 39.12 milt and . 10% travel more than 73.24 m/h, what are the mean and standard deviation of vehicle speed? [Note: The resulting values should agree with those given in the cited article.) b. What is the probability that a randomly selected vehicle's speed is between 50 and 65 m/h? c. What is the probability that a randomly selected vehicle's speed exceeds the speed limit of 70 m/h?A stockroom currently has 30 components of a certain type, of which 8 were provided by supplier 1, 10 by supplier 2, and 12 by supplier 3. Six of these are to be randomly selected for a particular assembly. Let X = the number of supplier 1's components selected, Y = the number of supplier 2's components selected, and p(x, y) denote the joint pmf of X and Y. a. What is p(3, 2)? [Hint: Each sample of size 6 is equally likely to be selected. Therefore, p(3, 2) = (number of outcomes with X = 3 and Y = 2)/(total number of outcomes). Now use the product rule for counting to obtain the numerator and denominator.] b. Using the logic of part (a), obtain p(x, y). (This can be thought of as a multivariate hypergeometric distribution-sampling without replacement from a finite population consisting of more than two categories.)Refer to Exercise 1 and answer the following questions: a. Given that X = 1, determine the conditional pmf of Y-i.e., Prix(0 | 1), Prix(1 | 1), and Prix(2 | 1). b. Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island? c. Use the result of part (b) to calculate the conditional probability P(Y s 1 [ X s 2). d. Given that two hoses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island? Reference Exercise 1 A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. - 2 p(x, y) 0 2 .10 .04 .02 N - O 08 .20 06 .06 14 30 a. What is P(X = 1 and Y = 1)? b. Compute P(X = 1 and Y = 1). c. Give a word description of the event (X #0 and Y # 0), and compute the probability of this event. d. Compute the marginal pmf of X and of Y. Using px(x), what is P(X = 1)? e. Are X and Y independent rv's? Explain
