Page 79 - Applied statistics and probability for engineers

Section 2-8/Random Variables 57

and chlorinated compounds—instead of having to use a single and 0.7, respectively. What is the probability that a visitor is from
test for each pollutant. The makers of the test claim that it can a search site given that the blue ad was viewed?
detect high levels of organic pollutants with 99.7% accuracy, 2-178. Suppose that a patient is selected randomly from those
volatile solvents with 99.95% accuracy, and chlorinated com- described in Exercise 2-98. Let A denote the event that the patient
pounds with 89.7% accuracy. If a pollutant is not present, the is in group 1, and let B denote the event that there is no progres-
test does not signal. Samples are prepared for the calibration of sion. Determine the following probabilities:
the test and 60% of them are contaminated with organic pollut- (a) P (B) (b) P (B | A) (c) P (A | B)
ants, 27% with volatile solvents, and 13% with traces of chlo- 2-179. An e-mail ilter is planned to separate valid e-mails
rinated compounds. A test sample is selected randomly. from spam. The word free occurs in 60% of the spam mes-
(a) What is the probability that the test will signal? sages and only 4% of the valid messages. Also, 20% of the
(b) If the test signals, what is the probability that chlorinated messages are spam. Determine the following probabilities:
compounds are present? (a) The message contains free.
2-174. Consider the endothermic reactions in Exercise 2-50. (b) The message is spam given that it contains free.
Use Bayes’ theorem to calculate the probability that a reac- (c) The message is valid given that it does not contain free.
tion's inal temperature is 271 K or less given that the heat
absorbed is above target. 2-180. A recreational equipment supplier inds that among orders
2-175. Consider the hospital emergency room data in that include tents, 40% also include sleeping mats. Only 5% of
Example 2-8. Use Bayes’ theorem to calculate the probability orders that do not include tents do include sleeping mats. Also,
that a person visits hospital 4 given they are LWBS. 20% of orders include tents. Determine the following probabilities:
2-176. Consider the well failure data in Exercise 2-53. Use (a) The order includes sleeping mats.
Bayes’ theorem to calculate the probability that a randomly (b) The order includes a tent given it includes sleeping mats.
selected well is in the gneiss group given that the well has 2-181. The probabilities of poor print quality given no printer
failed. problem, misaligned paper, high ink viscosity, or printer-head
2-177. Two Web colors are used for a site advertisement. If a site debris are 0, 0.3, 0.4, and 0.6, respectively. The probabilities
visitor arrives from an afiliate, the probabilities of the blue or green of no printer problem, misaligned paper, high ink viscosity, or
colors being used in the advertisement are 0.8 and 0.2, respectively. printer-head debris are 0.8, 0.02, 0.08, and 0.1, respectively.
If the site visitor arrives from a search site, the probabilities of blue (a) Determine the probability of high ink viscosity given poor
and green colors in the advertisement are 0.4 and 0.6, respectively. print quality.
The proportions of visitors from afiliates and search sites are 0.3 (b) Given poor print quality, what problem is most likely?

2-8 Random Variables

We often summarize the outcome from a random experiment by a simple number. In many
of the examples of random experiments that we have considered, the sample space has been
a description of possible outcomes. In some cases, descriptions of outcomes are suficient,
but in other cases, it is useful to associate a number with each outcome in the sample space.
Because the particular outcome of the experiment is not known in advance, the resulting value
of our variable is not known in advance. For this reason, the variable that associates a number
with the outcome of a random experiment is referred to as a random variable.
Random
Variable A random variable is a function that assigns a real number to each outcome in the
sample space of a random experiment.

Notation is used to distinguish between a random variable and the real number.

Notation
A random variable is denoted by an uppercase letter such as X. After an experiment
is conducted, the measured value of the random variable is denoted by a lowercase
letter such as x = 70 milliamperes.

Sometimes a measurement (such as current in a copper wire or length of a machined
part) can assume any value in an interval of real numbers (at least theoretically). Then
arbitrary precision in the measurement is possible. Of course, in practice, we might round