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Assume that each of your calls to a popular radio station has a probability of 002 of connecting, that is, of not obtaining a busy signal Assume that your calls are independent (a) What is the probability that your rst call that connects is your tenth call (b) What is the probability that it requires more than ve calls for you to connect (c) What is the mean number of calls needed to connect 3-76 In Exercise 3-70, recall that a particularly long traf c light on your morning commute is green 20% of the time that you approach it Assume that each morning represents an independent trial (a) What is the probability that the rst morning that the light is green is the fourth morning that you approach it (b) What is the probability that the light is not green for 10 consecutive mornings 3-77.

A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE) The probability of a computer failing in a day is 0005, and the computers fail independently Computers are repaired in the evening and each day is an independent trial (a) What is the probability that all eight computers fail in a day (b) What is the mean number of days until a speci c computer fails (c) What is the mean number of days until all eight computers fail in the same day 3-78 In Exercise 3-66, recall that 20 parts are checked each hour and that X denotes the number of parts in the sample of 20 that require rework.

28. Consider the multiple transmission attack against NTRU discussed in Section 6.7.2. Suppose that each r, is a polynomial of degree four or less and = -1 -2 2 x4

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(a) If the percentage of parts that require rework remains at 1%, what is the probability that hour 10 is the rst sample at which X exceeds 1 (b) If the rework percentage increases to 4%, what is the probability that hour 10 is the rst sample at which X exceeds 1.

(c) If the rework percentage increases to 4%, what is the expected number of hours until X exceeds 1 3-79 Consider a sequence of independent Bernoulli trials with p 02 (a) What is the expected number of trials to obtain the rst success (b) After the eighth success occurs, what is the expected number of trials to obtain the ninth success 3-80 Show that the probability density function of a negative binomial random variable equals the probability density function of a geometric random variable when r 1 Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for geometric random variable when r 1 3-81 Suppose that X is a negative binomial random variable with p 02 and r 4.

r 2 ( z )Q(2)-

Determine the following: (a) E1X 2 (b) P1X 202 (c) P1X 192 (d) P1X 212 (e) The most likely value for X 3-82 The probability is 06 that a calibration of a transducer in an electronic instrument conforms to speci cations for the measurement system Assume the calibration attempts are independent What is the probability that at most three calibration attempts are required to meet the speci cations for the measurement system 3-83 An electronic scale in an automated lling operation stops the manufacturing line after three underweight packages are detected Suppose that the probability of an underweight package is 0001 and each ll is independent (a) What is the mean number of lls before the line is stopped (b) What is the standard deviation of the number of lls before the line is stopped 3-84.

A fault-tolerant system that processes transactions for a nancial services rm uses three separate computers If the operating computer fails, one of the two spares can be immediately switched online After the second computer fails, the last computer can be immediately switched online Assume that the probability of a failure during any transaction is 10 8 and that the transactions can be considered to be independent events (a) What is the mean number of transactions before all computers have failed (b) What is the variance of the number of transactions before all computers have failed 3-85 Derive the expressions for the mean and variance of a geometric random variable with parameter p (Formulas for in nite series are required).

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