Devore Probability Statistics Engineering Sciences 8th txtbk

4.2 Cumulative Distribution Functions and Expected Values 143c. What is the probability that the lecture continues beyondthe hour for between 60 and 90 sec?d. What is the probability that the lecture continues for atleast 90 sec beyond the end of the hour?6. The actual tracking weight of a stereo cartridge that is set totrack at 3 g on a particular changer can be regarded as a continuousrv X with pdff(x) 5 e k[1 2 (x 2 3)2 ] 2 # x # 40 otherwisea. Sketch the graph of f(x).b. Find the value of k.c. What is the probability that the actual tracking weight isgreater than the prescribed weight?d. What is the probability that the actual weight is within.25 g of the prescribed weight?e. What is the probability that the actual weight differs fromthe prescribed weight by more than .5 g?7. The time X (min) for a lab assistant to prepare the equipmentfor a certain experiment is believed to have a uniform distributionwith A 5 25 and B 5 35.a. Determine the pdf of X and sketch the correspondingdensity curve.b. What is the probability that preparation time exceeds33 min?c. What is the probability that preparation time is within2min of the mean time? [Hint: Identify m from the graphof f(x).]d. For any a such that 25 , a , a 1 2 , 35, what is theprobability that preparation time is between a anda 1 2 min?8. In commuting to work, a professor must first get on a busnear her house and then transfer to a second bus. If the waitingtime (in minutes) at each stop has a uniform distributionwith A 5 0 and B 5 5, then it can be shown that the totalwaiting time Y has the pdf125 y 0 # y , 5f(y) 5 e25 2 125 y 5 # y # 100 y , 0 or y . 10a. Sketch a graph of the pdf of Y.`32`b. Verify that f(y) dy 5 1.c. What is the probability that total waiting time is at most3min?d. What is the probability that total waiting time is at most8 min?e. What is the probability that total waiting time is between3 and 8 min?f. What is the probability that total waiting time is eitherless than 2 min or more than 6 min?9. Consider again the pdf of X 5 time headway given inExample 4.5. What is the probability that time headway isa. At most 6 sec?b. More than 6 sec? At least 6 sec?c. Between 5 and 6 sec?10. A family of pdf’s that has been used to approximate the distributionof income, city population size, and size of firms isthe Pareto family. The family has two parameters, k and u,both . 0, and the pdf isk # u kf(x; k, u) 5 u x k11x $ u0 x , ua. Sketch the graph of f(x; k, u) .b. Verify that the total area under the graph equals 1.c. If the rv X has pdf f(x; k, u) , for any fixed b . u, obtainan expression for P(X # b) .d. For u , a , b, obtain an expression for the probabilityP(a # X # b).4.2 Cumulative Distribution Functionsand Expected ValuesSeveral of the most important concepts introduced in the study of discrete distributionsalso play an important role for continuous distributions. Definitions analogousto those in Chapter 3 involve replacing summation by integration.The Cumulative Distribution FunctionThe cumulative distribution function (cdf) F(x) for a discrete rv X gives, for anyspecified number x, the probability P(X # x) . It is obtained by summing the pmfp(y) over all possible values y satisfying y # x. The cdf of a continuous rv gives thesame probabilities P(X # x) and is obtained by integrating the pdf f(y) between thelimits 2` and x.Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.