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p87_exercises-3_6

# p87_exercises-3_6 - 3—7 GEOMETRIC AND NEGATIVE BINOMIAL...

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Unformatted text preview: 3—7 GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS 87 ~ . -' that 20 malicious software instances are reported. . m e that the malicious sources can be assumed to be inde- nt. J‘ o t is the probability at least one instance is “Klez”? .2 -.t is the probability that three or more instances are '“Klez”? ‘i‘ u t is the mean and standard deviation of the number of ' ez” instances among the 20 reported? ‘1 Heart failure is due to either natural occurrences (87%) .,-.ide factors (13%). Outside factors are related to induced Qzuces or foreign objects. Natural occurrences are caused j‘w 'al blockage, disease, and infection. Suppose that 20 pa- will visit an emergency room with heart failure. Assume , s of heart failure between individuals are independent. j. . t is the probability that three individuals have condi- i'ms caused by outside factors? *.' at is the probability that three or more individuals have conditions caused by outside factors? 7" at is the mean and standard deviation of the number of ' ‘ 'duals with conditions caused by outside factors? A computer system uses passwords that are exactly in 2 cters and each character is one of the 26 letters (a—z) integers (0—9). Suppose there are 10000 users on the sys- with unique passwords. A hacker randomly selects (with ‘ ent) one billion passwords from the potential set in i' iseconds before security software closes the unautho- ‘j' access. "“1“ I t is the distribution of the number of user passwords lected by the hacker? ‘ at is the probability that no user passwords are se- - ted? ” s t is the mean and variance of the number of user pass- x" rds selected? is . A statistical process control chart example. Samples 7.1 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more than three standard deviations. (a) If the percentage of parts that require rework remains at 1%, what is .the probability that X exceeds its mean by more than three standard deviations? (b) If the rework percentage increases to 4%, what is the probability thatX exceeds 1? (c) If the rework percentage increases to 4%, what is the probability thatX exceeds 1 in at least one of the next ﬁve hours of samples? 3—79. Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a ﬂight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently. (a) What is the probability that every passenger who shows up can take the ﬂight? (b) What is the probability that the ﬂight departs with empty seats? 3—80. This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 cus- tomer orders to ﬁll. Each order requires one component part that is purchased from a supplier. However, typically, 2% of the components are identiﬁed as defective, and the compo- nents can be assumed to be independent. (a) If the manufacturer stocks 100 components, what is the probability that the 100 orders can be ﬁlled without reordering components? (b) If the manufacturer stocks 102 components, what is the probability that the 100 orders can be ﬁlled without reordering components? (c) If the manufacturer stocks 105 components, what is the probability that the 100 orders can be ﬁlled. without reordering components? GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS .1 Geometric Distribution Consider a random experiment that is closely related to the one used in the deﬁnition of a binomial distribution. Again, assume a series of Bernoulli trials (independent trials with con- stant probability p of a success on each trial). However, instead of a ﬁxed number of trials, trials are conducted until a success is obtained. Let the random variable X denote the number of trials until the ﬁrst success. In Example 3-5, successive wafers are analyzed until a large particle is detected. Then, X is the number of wafers analyzed. In the transmission of bits, X might be the number of bits transmitted until an error occurs. 'LE 3—20 The probability that a bit transmitted through a digital transmission channel is received in error is 0.1. i 'tal Channel 1;? Assume the transmissions are independent events, and let the random variable X denote the number of 7;: . bits transmitted until the ﬁrst error. ...
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