Unformatted text preview: 86 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS (h)A ﬁlling operation attempts to ﬁll detergent packages to
the advertised weight. Let X denote the number of deter
gent packages that are underﬁlled. (i) Errors in a digital communication channel occur in bursts
that affect several consecutive bits. Let X denote the num
ber of bits in error in a transmission of 100,000 bits. 0) LetX denote the number of surface ﬂaws in a large coil of
galvanized steel. 3—62. Let X be a binomial random variable with p = 0.2
and n = 20. Use the binomial table in Appendix A to deter
mine the following probabilities. (a) P(X s 3) (b) P(X > 10) (c) P(X: 6) (d) P(6 5X5 11) 3—63. Let X be a binomial random variable with p = 0.1
and n = 10. Calculate the following probabilities from the bi
nomial probability mass function and also from the binomial
table in Appendix A and compare results. (a) P(XS 2) (b) P(X> 8) (c) P(X = 4) (d) P(S s X S 7) 3—64. The random variable X has a binomial distribution
with n = 10 and p = 0.5. Determine the following proba
bilities: (a) P(X= 5) (b) P(XS 2)
(c) P(Xa 9) (d) P(3 s X< 5)
3—65. The random variable X has a binomial distribution with n = 10 and p = 0.01. Determine the following proba
bilities. (a) P(X = 5) (b) P(XS 2) (c) P(X 2 9) (d) P(3 s X < 5) 3—66. The random variable X has a binomial distribution with
n = 10 and p = 0.5. Sketch the probability mass function of X.
(a) What value of X is most likely? (b) What value(s) of X is(are) least likely? 3—67. Sketch the probability mass function of a binomial
distribution with n = 10 and p = 0.01 and comment on the
shape of the distribution. (a) What value of X is most likely? (b) What value of X is least likely? 3—68. Determine the cumulative distribution function of a
binomial random variable with n = 3 and p = 1/2. 3—69. Determine the cumulative distribution function of a
binomial random variable with n = 3 and p = 1 / 4. 32—70. An electronic product contains 40 integrated circuits.
The probability that any integrated circuit is defective is 0.01,
and the integrated circuits are independent. The product oper
ates only if there are no defective integrated circuits. What is
the probability that the product operates? 3—7 1. The phone lines to an airline reservation system are occupied 40% of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline. (a) What is the probability that for exactly three calls the lines
are occupied? (b) What is the probability that for at least one call the lin
are not occupied? (c) What is the expected number of calls in which the lin
are all occupied? question. (a) What is the probability that the student answers more
20 questions correctly? (b) What is the probability the student answers less than .
questions correctly? 3—73. A particularly long trafﬁc light on your morning com mute is green 20% of the time that you approach it. Assume. that each morning represents an independent trial. (3) Over ﬁve mornings, what is the probability that the light is
green on exactly one day? (b) Over 20 mornings, what is the probability that the light is
green on exactly four days? (0) Over 20 mornings, what is the probability that the light is
green on more than four days? 3—74. Samples of rejuvenated mitochondria are mutated
(defective) in 1% of cases. Suppose 15 samples are studied,
and they can be considered to be independent for mutation.‘
Determine the following probabilities. The binomial table in
Appendix A can help. (a) No samples are mutated. (b) At most one sample is mutated. (c) More than half the samples are mutated. 3—75. An article in Information Security Technical Report,
“Malicious software—past, present and future,” (2004, Vol. 9, '
pp. 6—18) provided the following data on the top ten malicious
software instances for 2002. The clear leader in the number of
registered incidences for the year 2002 was the Internet worm
“Klez, and it is still one of the most widespread threats. This
virus was ﬁrst detected on 26 October 2001, and it has held the
top spot among malicious software for the longest period in
the history of virology. 61 22% l , IWormKlez 2 IWorm.Lentin 20.52%
3 I—Worm.Tanatos 2.09%
4 IWorm.BadtIansII 1.3 1%
5 ' Macro.Word97.Thus , 1.19%
6 I—Worm.Hybris 0.60%
7 I.Worm.Bridex , 0.32%
8 IWorm.Magistr 0.30%
9 Win95.CIH 0.27%
10 IWorm.Sircam 0.24% The 10 most widespread malicious programs for 2002
(Soume—Kasperslty Labs). ...
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 Spring '08
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