Page 128 - Applied statistics and probability for engineers

106 Chapter 3/Discrete Random Variables and Probability Distributions

3-209. Saguaro cacti are large cacti indigenous to the (c) Area of a region such that the probability of at least two
southwestern United States and Mexico. Assume that the cacti in the region is 0.9.
number of saguaro cacti in a region follows a Poisson distri- 3-210. Suppose that 50 sites on a patient might contain lesions.
bution with a mean of 280 per square kilometer. Determine A biopsy selects 8 sites randomly (without replacement). What
the following: is the minimum number of sites with lesions so that the prob-
(a) Mean number of cacti per 10,000 square meters. ability of at least one selected site contains lesions is greater
(b) Probability of no cacti in 10,000 square meters. than or equal to 0.95? Rework for greater than or equal to 0.99.

Mind-Expanding Exercises

3-211. Derive the convergence results used to obtain a least one nonconforming item in the sample is at least 0.90?
Poisson distribution as the limit of a binomial distribution. Assume that the binomial approximation to the hypergeo-
3-212. Show that the function f x ( ) in Example 3-5 satis- metric distribution is adequate.
ies the properties of a probability mass function by sum- 3-217. A company performs inspection on shipments from
ming the ininite series. suppliers to detect nonconforming products. The company’s
3-213. Derive the formula for the mean and standard devi- policy is to use a sample size that is always 10% of the lot size.
ation of a discrete uniform random variable over the range Comment on the effectiveness of this policy as a general rule
of integers a, a +1 ,… b. for all sizes of lots.
3-214. Derive the expression for the variance of a geomet- 3-218. A manufacturer stocks components obtained from
ric random variable with parameter p. a supplier. Suppose that 2% of the components are defec-
3-215. An air light can carry 120 passengers. A passenger tive and that the defective components occur independently.
with a reserved seat arrives for the light with probability How many components must the manufacturer have in
0.95. Assume that the passengers behave independently. stock so that the probability that 100 orders can be com-
(Use of computer software is expected.) pleted without reordering components is at least 0.95?
(a) What is the minimum number of seats the airline should 3-219. A large bakery can produce rolls in lots of either 0,
reserve for the probability of a full light to be at least 0.90? 1000, 2000, or 3000 per day. The production cost per item
(b) What is the maximum number of seats the airline should is $0.10. The demand varies randomly according to the fol-
reserve for the probability that more passengers arrive lowing distribution:
than the light can seat to be less than 0.10?
(c) Discuss some reasonable policies the airline could use Demand for rolls 0 1000 2000 3000
to reserve seats based on these probabilities. Probability of demand 0.3 0.2 0.3 0.2
3-216. A company performs inspection on shipments from Every roll for which there is a demand is sold for $0.30.
suppliers to detect nonconforming products. Assume that a Every roll for which there is no demand is sold in a second-
lot contains 1000 items and 1% are nonconforming. What ary market for $0.05. How many rolls should the bakery
sample size is needed so that the probability of choosing at produce each day to maximize the mean proit?

Important Terms and Concepts

Bernoulli trial Finite population correction Negative binomial Standard deviation-
Binomial distribution factor distribution discrete random variable
Cumulative distribution Geometric distribution Poisson distribution Variance—discrete
function-discrete random Hypergeometric distribution Poisson process random variable
variable Lack of memory property- Probability distribution-
Discrete uniform distribution discrete random variable discrete random variable
Expected value Mean—discrete random Probability mass
of a function of a discrete variable function
random variable