Page 89 - Applied statistics and probability for engineers

Section 3-2/Probability Distributions and Probability Mass Functions 67

Example 3-3 Deine the random variable X to be the number of contamination particles on a wafer in semicon-

ductor manufacturing. Although wafers possess a number of characteristics, the random variable X
summarizes the wafer only in terms of the number of particles.
The possible values of X are integers from zero up to some large value that represents the maximum number of par-
ticles that can be found on one of the wafers. If this maximum number is large, we might simply assume that the range
of X is the set of integers from zero to ini nity.

Note that more than one random variable can be deined on a sample space. In Example 3-3, we

might also deine the random variable Y to be the number of chips from a wafer that fails the i nal test.
Exercises FOR SECTION 3-1

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
For each of the following exercises, determine the range (pos- 3-8. The random variable is the number of computer clock
sible values) of the random variable. cycles required to complete a selected arithmetic calculation.
3-1. The random variable is the number of nonconform- 3-9. An order for an automobile can select the base model
ing solder connections on a printed circuit board with 1000 or add any number of 15 options. The random variable is the
connections. number of options selected in an order.
3-2. In a voice communication system with 50 lines, the 3-10. Wood paneling can be ordered in thicknesses of
random variable is the number of lines in use at a particular time. 1 8, 1 4, or 3 8 inch. The random variable is the total thickness
3-3. An electronic scale that displays weights to the near- of paneling in two orders.
est pound is used to weigh packages. The display shows only 3-11. A group of 10,000 people are tested for a gene called

ive digits. Any weight greater than the display can indicate is Ii 202 that has been found to increase the risk for lupus. The
shown as 99999. The random variable is the displayed weight. random variable is the number of people who carry the gene.
3-4. A batch of 500 machined parts contains 10 that do not 3-12. In an acid-base titration, the milliliters of base that
conform to customer requirements. The random variable is the are needed to reach equivalence are measured to the nearest

number of parts in a sample of ive parts that do not conform to milliliter between 0.1 and 0.15 liters (inclusive).
customer requirements. 3-13. The number of mutations in a nucleotide sequence
3-5. A batch of 500 machined parts contains 10 that do of length 40,000 in a DNA strand after exposure to radiation is
not conform to customer requirements. Parts are selected suc- measured. Each nucleotide may be mutated.
cessively, without replacement, until a nonconforming part is 3-14. A healthcare provider schedules 30 minutes for each
obtained. The random variable is the number of parts selected. patient’s visit, but some visits require extra time. The random
3-6. The random variable is the moisture content of a lot variable is the number of patients treated in an eight-hour day.
of raw material, measured to the nearest percentage point. 3-15. A Web site contains 100 interconnected pages. The ran-

3-7. The random variable is the number of surface laws in a dom variable is the number of unique pages viewed by a visitor
large coil of galvanized steel. to the Web site.

3-2 Probability Distributions and Probability
Mass Functions

Random variables are so important in random experiments that sometimes we essentially
ignore the original sample space of the experiment and focus on the probability distribution
of the random variable. For example, in Example 3-1, our analysis might focus exclusively on
the integers {0, 1, . . . , 48} in the range of X. In Example 3-2, we might summarize the ran-
dom experiment in terms of the three possible values of X, namely {0, 1, 2}. In this manner, a
random variable can simplify the description and analysis of a random experiment.
The probability distribution of a random variable X is a description of the probabilities
associated with the possible values of X. For a discrete random variable, the distribution is
often specii ed by just a list of the possible values along with the probability of each. In some
cases, it is convenient to express the probability in terms of a formula.