Page 118 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 96

96 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

MIND-EXPANDING EXERCISES

3-130. Derive the mean and variance of a hypergeo- 3-135. Surface flaws in automobile exterior panels
metric random variable (difficult exercise). follow a Poisson distribution with a mean of 0.1 flaw per
3-131. Show that the function f(x) in Example 3-5 panel. If 100 panels are checked, what is the probability
satisfies the properties of a probability mass function by that fewer than five panels have any flaws?
summing the infinite series. 3-136. A large bakery can produce rolls in lots of ei-
3-132. Derive the formula for the mean and standard ther 0, 1000, 2000, or 3000 per day. The production cost
deviation of a discrete uniform random variable over the per item is $0.10. The demand varies randomly accord-
range of integers a, a 1, p , b . ing to the following distribution:
3-133. A company performs inspection on shipments demand for rolls 0 1000 2000 3000
from suppliers in order to defect nonconforming prod- probability of demand 0.3 0.2 0.3 0.2
ucts. Assume a lot contains 1000 items and 1% are
nonconforming. What sample size is needed so that the Every roll for which there is a demand is sold for $0.30.
probability of choosing at least one nonconforming item Every roll for which there is no demand is sold in a sec-
in the sample is at least 0.90? Assume the binomial ondary market for $0.05. How many rolls should the
approximation to the hypergeometric distribution is bakery produce each day to maximize the mean profit?
adequate. 3-137. A manufacturer stocks components obtained
3-134. A company performs inspection on shipments from a supplier. Suppose that 2% of the components are
from suppliers in order to detect nonconforming prod- defective and that the defective components occur inde-
ucts. The company’s policy is to use a sample size that is pendently. How many components must the manufacturer
always 10% of the lot size. Comment on the effective- have in stock so that the probability that 100 orders can be
ness of this policy as a general rule for all sizes of lots. completed without reordering components is at least 0.95?

IMPORTANT TERMS AND CONCEPTS
In the E-book, click on any Expected value of a Mean-discrete random Probability mass
term or concept below to function of a random variable function
go to that subject. variable Mean-function of a Standard deviation-
Bernoulli trial Finite population discrete random discrete random
Binomial distribution correction factor variable variable
Cumulative probability Geometric distribution Negative binomial Variance-discrete
distribution function- Hypergeometric distri- distribution random variable
discrete random bution Poisson distribution
variable Lack of memory Poisson process
Discrete uniform distri- property-discrete Probability distribution-
bution random variable discrete random
variable