Page 164 - Applied statistics and probability for engineers

142 Chapter 4/Continuous Random Variables and Probability Distributions

where X is gamma with λ = .5 and r = 10 . In software, we use the gamma inverse cumulative probability function and
0
set the shape parameter to 10, the scale parameter to 0.5, and the probability to 0.95. The solution is
(
P X ≤ 31 . ) = .95
0
41
Practical Interpretation: Based on this result, a schedule that allows 31.41 hours to prepare 10 slides should be met
95% of the time.
Furthermore, the chi-squared distribution is a special case of the gamma distribution
/
in which λ = 1 2 and r equals one of the values 1/2, 1, 3/2, 2, . . . . This distribution is used
extensively in interval estimation and tests of hypotheses that are discussed in subsequent
chapters. The chi-squared distribution is discussed in Chapter 7.

EXERCISES FOR SECTION 4-9

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-137. Use the properties of the gamma function to evalu- (b) What is the standard deviation of the time until a packet is
ate the following: formed?
/
/
(a) Γ( ) 6 (b) Γ(5 2 ) (c) Γ(9 2 ) (c) What is the probability that a packet is formed in less than
10 seconds?
3
4-138. Given the probability density function f x ( ) = .01 (d) What is the probability that a packet is formed in less than
0
.
2
x e − 0 01 x / Γ( 3) , determine the mean and variance of the
i ve seconds?
distribution. 4-143. Errors caused by contamination on optical disks
4-139. Calls to a telephone system follow a Poisson dis- occur at the rate of one error every 10 bits. Assume that the
5
tribution with a mean of ive calls per minute. errors follow a Poisson distribution.

(a) What is the name applied to the distribution and parameter

values of the time until the 10th call? (a) What is the mean number of bits until ive errors occur?
(b) What is the mean time until the 10th call? (b) What is the standard deviation of the number of bits until

(c) What is the mean time between the 9th and 10th calls? ive errors occur?
(d) What is the probability that exactly four calls occur within (c) The error-correcting code might be ineffective if there are
5
one minute? three or more errors within 10 bits. What is the probability
(e) If 10 separate one-minute intervals are chosen, what is the of this event?
probability that all intervals contain more than two calls? 4-144. Calls to the help line of a large computer distribu-
tor follow a Poisson distribution with a mean of 20 calls per
4-140. Raw materials are studied for contamination. Sup- minute. Determine the following:
pose that the number of particles of contamination per pound (a) Mean time until the one-hundredth call
of material is a Poisson random variable with a mean of 0.01
particle per pound. (b) Mean time between call numbers 50 and 80
(a) What is the expected number of pounds of material required (c) Probability that three or more calls occur within 15 seconds
to obtain 15 particles of contamination? 4-145. The time between arrivals of customers at an auto-
(b) What is the standard deviation of the pounds of materials matic teller machine is an exponential random variable with a
required to obtain 15 particles of contamination? mean of i ve minutes.
(a) What is the probability that more than three customers
4-141. The time between failures of a laser in a cytogenics
machine is exponentially distributed with a mean of 25,000 hours. arrive in 10 minutes?
(a) What is the expected time until the second failure? (b) What is the probability that the time until the i fth customer
(b) What is the probability that the time until the third failure arrives is less than 15 minutes?
=
r
exceeds 50,000 hours? 4-146. Use integration by parts to show that Γ( ) (r − ) 1
Γ(r − ).
1
4-142. In a data communication system, several messages
(

that arrive at a node are bundled into a packet before they are 4-147. Show that the gamma density function f x, ,r) inte-
λ
transmitted over the network. Assume that the messages arrive grates to 1.
at the node according to a Poisson process with τ = 30 mes- 4-148. Use the result for the gamma distribution to determine
sages per minute. Five messages are used to form a packet. the mean and variance of a chi-square distribution with r = 7 2.
(a) What is the mean time until a packet is formed, that is, until 4-149. Patients arrive at a hospital emergency department
ive messages have arrived at the node? according to a Poisson process with a mean of 6.5 per hour.