Page 107 - Applied statistics and probability for engineers

Section 3-6/Binomial Distribution 85

3-94. The random variable X has a binomial distribution (a) No samples are mutated.
with n = 10 and p = 0.5. Determine the following probabilities: (b) At most one sample is mutated.
(
(
(a) P X = ) 5 (b) P X ≤ ) 2 (c) More than half the samples are mutated.
( (d) P 3 ≤ X < 5) 3-105. An article in Information Security Technical Report
(
(c) P X ≥ ) 9
3-95. The random variable X has a binomial distribution [“Malicious Software—Past, Present and Future” (2004, Vol. 9,
pp. 6–18)] provided the following data on the top 10 malicious
with n = 10 and p = 0.01. Determine the following probabilities.
(
(
(a) P X = ) 5 (b) P X ≤ ) 2 software instances for 2002. The clear leader in the number of
(
( (d) P 3 ≤ X < 5) registered incidences for the year 2002 was the Internet worm
(c) P X ≥ ) 9 “Klez,” and it is still one of the most widespread threats. This
3-96. The random variable X has a binomial distribution with virus was irst detected on 26 October 2001, and it has held the
n = 10 and p = 0.5. Sketch the probability mass function of X. top spot among malicious software for the longest period in the
(a) What value of X is most likely? history of virology.
(b) What value(s) of X is(are) least likely? The 10 most widespread malicious programs for 2002
3-97. Sketch the probability mass function of a binomial dis-
tribution with n = 10 and p = 0.01 and comment on the shape Place Name % Instances
of the distribution. 1 I-Worm.Klez 61.22%
(a) What value of X is most likely?
2 I-Worm.Lentin 20.52%
(b) What value of X is least likely?
3-98. Determine the cumulative distribution function of a 3 I-Worm.Tanatos 2.09%
binomial random variable with n = 3 and p = 1 2. 4 I-Worm.BadtransII 1.31%
3-99. Determine the cumulative distribution function of a 5 Macro.Word97.Thus 1.19%
binomial random variable with n = 3 and p = 1 4. 6 I-Worm.Hybris 0.60%
3-100. An electronic product contains 40 integrated cir-
cuits. The probability that any integrated circuit is defective is 7 I-Worm.Bridex 0.32%
0.01, and the integrated circuits are independent. The product 8 I-Worm.Magistr 0.30%
operates only if there are no defective integrated circuits. What
9 Win95.CIH 0.27%
is the probability that the product operates?
10 I-Worm.Sircam 0.24%
3-101. The phone lines to an airline reservation system are
occupied 40% of the time. Assume that the events that the lines (Source: Kaspersky Labs).
are occupied on successive calls are independent. Assume that Suppose that 20 malicious software instances are reported.
10 calls are placed to the airline. Assume that the malicious sources can be assumed to be
(a) What is the probability that for exactly three calls, the lines independent.
are occupied? (a) What is the probability that at least one instance is “Klez?”
(b) What is the probability that for at least one call, the lines (b) What is the probability that three or more instances are “Klez?”
are not occupied? (c) What are the mean and standard deviation of the number of
(c) What is the expected number of calls in which the lines are “Klez” instances among the 20 reported?
all occupied? 3-106. Heart failure is due to either natural occurrences
3-102. A multiple-choice test contains 25 questions, each (87%) or outside factors (13%). Outside factors are related to
with four answers. Assume that a student just guesses on each induced substances or foreign objects. Natural occurrences are
question. caused by arterial blockage, disease, and infection. Suppose
(a) What is the probability that the student answers more than 20 that 20 patients will visit an emergency room with heart fail-
questions correctly? ure. Assume that causes of heart failure for the individuals are
(b) What is the probability that the student answers fewer than 5 independent.
questions correctly? (a) What is the probability that three individuals have condi-
3-103. A particularly long trafic light on your morning com- tions caused by outside factors?
mute is green 20% of the time that you approach it. Assume that (b) What is the probability that three or more individuals have
each morning represents an independent trial. conditions caused by outside factors?
(a) Over 5 mornings, what is the probability that the light is (c) What are the mean and standard deviation of the number
green on exactly one day? of individuals with conditions caused by outside factors?
(b) Over 20 mornings, what is the probability that the light is 3-107. A computer system uses passwords that are exactly six
green on exactly four days? characters and each character is one of the 26 letters (a–z) or
(c) Over 20 mornings, what is the probability that the light is 10 integers (0–9). Suppose that 10,000 users of the system have
green on more than four days? unique passwords. A hacker randomly selects (with replace-
3-104. Samples of rejuvenated mitochondria are mutated ment) one billion passwords from the potential set, and a match
(defective) in 1% of cases. Suppose that 15 samples are studied to a user’s password is called a hit.
and can be considered to be independent for mutation. Deter- (a) What is the distribution of the number of hits?
mine the following probabilities. The binomial table in Appen- (b) What is the probability of no hits?
dix A can help. (c) What are the mean and variance of the number of hits?