Page 117 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 95

3-9 POISSON DISTRIBUTION 95

of the number of keywords (used to help route messages) and 3-125. An installation technician for a specialized commu-
the type of message, either email or voice. Also, 70% of the nication system is dispatched to a city only when three or
messages arrive via email and the rest are voice. more orders have been placed. Suppose orders follow a
Poisson distribution with a mean of 0.25 per week for a city
number of keywords 0 1 2 3 4 with a population of 100,000 and suppose your city contains a
email 0.1 0.1 0.2 0.4 0.2 population of 800,000.
voice 0.3 0.4 0.2 0.1 0 (a) What is the probability that a technician is required after a
one-week period?
Determine the probability mass function of the number of (b) If you are the first one in the city to place an order, what is
keywords in a message. the probability that you have to wait more than two weeks
3-121. The random variable X has the following probability from the time you place your order until a technician is
distribution: dispatched?
3-126. From 500 customers, a major appliance manufac-
x 2 3 5 8 turer will randomly select a sample without replacement. The
probability 0.2 0.4 0.3 0.1 company estimates that 25% of the customers will provide
useful data. If this estimate is correct, what is the probability
Determine the following: mass function of the number of customers that will provide
(a) P1X 32 (b) P1X 2.52 useful data?
(c) P12.7 X 5.12 (d) E1X2 (a) Assume that the company samples 5 customers.
(e) V1X2 (b) Assume that the company samples 10 customers.
3-122. Determine the probability mass function for the ran- 3-127. It is suspected that some of the totes containing
dom variable with the following cumulative distribution chemicals purchased from a supplier exceed the moisture con-
function: tent target. Samples from 30 totes are to be tested for moisture
content. Assume that the totes are independent. Determine the
0 x 2 proportion of totes from the supplier that must exceed the
0.2 2 x 5.7 moisture content target so that the probability is 0.90 that at
F1x2 µ0.5 5.7 x 6.5 least one tote in the sample of 30 fails the test.
0.8 6.5 x 8.5 3-128. Messages arrive to a computer server according
1 8.5 x to a Poisson distribution with a mean rate of 10 per
hour. Determine the length of an interval of time such that
3-123. Each main bearing cap in an engine contains four the probability that no messages arrive during this interval
bolts. The bolts are selected at random, without replacement, is 0.90.
from a parts bin that contains 30 bolts from one supplier and Flaws occur in the interior of plastic used for auto-
70 bolts from another. 3-129.
mobiles according to a Poisson distribution with a mean of
(a) What is the probability that a main bearing cap contains 0.02 flaw per panel.
all bolts from the same supplier? (a) If 50 panels are inspected, what is the probability that
(b) What is the probability that exactly three bolts are from there are no flaws?
the same supplier? (b) What is the expected number of panels that need to be
3-124. Assume the number of errors along a magnetic inspected before a flaw is found?
recording surface is a Poisson random variable with a mean of (c) If 50 panels are inspected, what is the probability that the
one error every 10 5 bits. A sector of data consists of 4096 number of panels that have one or more flaws is less than
eight-bit bytes. or equal to 2?
(a) What is the probability of more than one error in a sector?
(b) What is the mean number of sectors until an error is found?