Page 113 - Applied statistics and probability for engineers

Section 3-7/Geometric and Negative Binomial Distributions 91

Example 3-25 Camera Flashes Consider the time to recharge the l ash in Example 3-25. The probability that a
camera passes the test is 0.8, and the cameras perform independently. What is the probability that the

third failure is obtained in ive or fewer tests?
Let X denote the number of cameras tested until three failures have been obtained. The requested probability is
(
P X ≤ ) 5 . Here X has a negative binomial distribution with p = 0 2 and r = 3. Therefore,
.
5
4 4⎞
(
P X ≤ ) = ∑ ⎛ ⎜ x − ⎞ 1 ⎟ 0 2 0 8) x−3 = 0 2. 3 + ⎛ ⎞ 3 0 2 0 8) + ⎛ ⎜ ⎟ 0 2 0 8 =. ( . ) 2 0 056
3
3
3
5
.
(
.
.
.
.
(
⎜ ⎟
x=3 ⎝ 2 ⎠ ⎝ ⎠ 2 ⎝ 2⎠
Exercises FOR SECTION 3-7
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
3-119. Suppose that the random variable X has a geometric (b) What is the probability that it requires more than i ve calls
distribution with p = 0 5. . Determine the following probabilities: for you to connect?
(
(
(
(a) P X = ) 1 (b) P X = ) 4 (c) P X = ) 8 (c) What is the mean number of calls needed to connect?
( ( 3-126. A player of a video game is confronted with a series
(d) P X ≤ ) 2 (e) P X > ) 2
of opponents and has an 80% probability of defeating each one.
3-120. Suppose that the random variable X has a geometric dis- Success with any opponent is independent of previous encoun-
tribution with a mean of 2.5. Determine the following probabilities: ters. Until defeated, the player continues to contest opponents.
(
(
(
(a) P X = ) 1 (b) P X = ) 4 (c) P X = ) 5 (a) What is the probability mass function of the number of
( (
(d) P X ≤ ) 3 (e) P X > ) 3 opponents contested in a game?
(b) What is the probability that a player defeats at least two
3-121. Consider a sequence of independent Bernoulli tri-
opponents in a game?
als with p = 0 2. .
(c) What is the expected number of opponents contested in a game?
(a) What is the expected number of trials to obtain the i rst success?
(d) What is the probability that a player contests four or more
(b) After the eighth success occurs, what is the expected number
opponents in a game?
of trials to obtain the ninth success?
3-122. Suppose that X is a negative binomial random (e) What is the expected number of game plays until a player
contests four or more opponents?
variable with p = 0 2. and r = 4. Determine the following:
(
(a) E X ( ) (b) P X = ) 3-127. Heart failure is due to either natural occurrences
20
(c) P X = ) (d) P X = ) (87%) or outside factors (13%). Outside factors are related to
(
(
19
21
induced substances or foreign objects. Natural occurrences are
(e) The most likely value for X caused by arterial blockage, disease, and infection. Assume
3-123. The probability of a successful optical align- that causes of heart failure for the individuals are independent.
ment in the assembly of an optical data storage product is 0.8. (a) What is the probability that the i rst patient with heart fail-
Assume that the trials are independent. ure who enters the emergency room has the condition due
(a) What is the probability that the i rst successful alignment to outside factors?
requires exactly four trials? (b) What is the probability that the third patient with heart fail-
(b) What is the probability that the i rst successful alignment ure who enters the emergency room is the irst one due to

requires at most four trials? outside factors?
(c) What is the probability that the i rst successful alignment (c) What is the mean number of heart failure patients with the con-
requires at least four trials? dition due to natural causes who enter the emergency room
3-124. In a clinical study, volunteers are tested for a gene before the irst patient with heart failure from outside factors?

that has been found to increase the risk for a disease. The prob- 3-128. A computer system uses passwords constructed
ability that a person carries the gene is 0.1. from the 26 letters (a–z) or 10 integers (0–9). Suppose that
(a) What is the probability that four or more people need to be 10,000 users of the system have unique passwords. A hacker
tested to detect two with the gene? randomly selects (with replacement) passwords from the
(b) What is the expected number of people to test to detect two potential set.
with the gene? (a) Suppose that 9900 users have unique six-character passwords
3-125. Assume that each of your calls to a popular radio and the hacker randomly selects six-character passwords.
station has a probability of 0.02 of connecting, that is, of not What are the mean and standard deviation of the number of
obtaining a busy signal. Assume that your calls are independent. attempts before the hacker selects a user password?

(a) What is the probability that your irst call that connects is (b) Suppose that 100 users have unique three-character passwords
your 10th call? and the hacker randomly selects three-character passwords.