c. What is the probability that the next car tuned has at least six cylinders? More than six cylinders?
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1 Exercises Section 3.2 [page 98] 11. An automobile service facility specializing in engine tune-ups knows that %&% of all tune-ups are done on four-cylinder automobiles, %!% on six-cylinder automobiles, and "&% on eight-cylinder automobiles. Let \œthe number of cylinders on the next car to be tuned. a. What is the pmf of \? b. Draw both a line graph and a probability histogram for the pmf of part a. c. What is the probability that the next car tuned has at least six cylinders? More than six cylinders?
2 12. Airlines sometimes overbook flights. Suppose that for a plane with &! seats, && passengers have tickets. Define the rv ] as the number of ticketed passengers who actually show up for the flight. The probability mass function of ] appears in the following table. C %& %' %( %) %* &! &" &# &$ &% && :ÐCÑ Þ!& Þ"! Þ"# Þ"% Þ#& Þ"( Þ!' Þ!& Þ!$ Þ!# Þ!" 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?
3 13. A mail-order computer business has six telephone lines. Let \ denote the number of lines in use at a specified time. Suppose the pmf of \ is as give in the accompanying table. B! " # $ % & ' :ÐBÑ Þ"! Þ"& Þ#! Þ#& Þ#! Þ!' Þ!% Calculate the probability of each of the following events. a. Ö at most three lines are in use b. Ö fewer than three lines are in use c. Ö at least three lines are in use d. Ö between two and five lines, inclusive, are in use e. Ö between two and four lines, inclusive, are not in use f. Ö at least four lines are not in use
4 15. Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives computer boards in lots of five. Two boards are selected from each lot for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair Ð"ß #Ñ represents the selection of boards " and # for inspection. a. List the ten different possible outcomes. b. Suppose that boards " and # are the only defective boards in a lot of five. Two boards are to be chosen at random. Define \ to be the number of defective boards observed among those inspected. Find the probability distribution of \. c. Let J ÐBÑ denote the cdf of \. First determine J Ð!Ñ œ T Ð\ Ÿ!Ñß JÐ"Ñand JÐ#Ñ; then obtain JÐBÑfor all other B.
5 16. some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area, $! % of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let \ denote the number among the four who have earthquake insurance. a. Find the probability distribution of \. [ Hint: Let W denote a homeowner who has insurance and J one who does not. Then one possible outcome is WJ WW, with probability ÐÞ$ÑÐÞ(ÑÐÞ$ÑÐÞ$Ñ and associated \ value $. There are "& other outcomes.] b. Draw the corresponding probability histogram. c. What is the most likely value for \? d. What is the probability that at least two of the four selected have earthquake insurance?
6 23. A consumer organization that evaluates new automobiles customarily reports the number of major defects in each car examined. Let \ denote the number of major defects in a randomly selected car of a certain type. The cdf of \ is as follows: Ú! B! Þ!'!ŸB " Þ"* "ŸB # Ý Þ$* #ŸB $ J ÐBÑ œ Û Þ'( $ŸB % Þ*# %ŸB & Ý Þ*( &ŸB ' Ü " ' Ÿ B Calculate the following probabilities directly from the cdf: a. :Ð#Ñ, that is, T Ð\ œ #Ñ b. TÐ\ $Ñ c. TÐ# Ÿ \ Ÿ &Ñ d. TÐ# \ &Ñ
7 24. An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let \œthe number of months between successive payments. The cdf of \ is given here. Ú! if B " Ý Þ$! if "ŸB $ Þ%! if $ŸB % J ÐBÑ œ Û Þ%& if %ŸB ' Ý Þ'! if 'ŸB "# Ü " if "# B a. What is the pmf of \? b. Using just the cdf, compute TÐ$ Ÿ \ Ÿ 'Ñ and TÐ% Ÿ \Ñ.
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