Page 102 - Applied statistics and probability for engineers

80 Chapter 3/Discrete Random Variables and Probability Distributions

3-88. The probability that data are entered incorrectly into a selected randomly and independently for each question. Let X

ield in a database is 0.005. A data entry form has 28 i elds, and denote the number of questions answered correctly. Does X

errors occur independently for each ield. The random variable have a discrete uniform distribution? Why or why not?
X is the number of ields on the form with an error. Does X 3-90. Consider the hospital data in Example 2-8. Suppose a

have a discrete uniform distribution? Why or why not? patient is selected randomly from the collection in the table. Let X
3-89. Each multiple-choice question on an exam has four denote the hospital number of the selected patient (either 1, 2, 3, or
choices. Suppose that there are 10 questions and the choice is 4). Does X have a discrete uniform distribution? Why or why not?

3-6 Binomial Distribution

Consider the following random experiments and random variables:
1. Flip a coin 10 times. Let X = number of heads obtained.
2. A worn machine tool produces 1% defective parts. Let X = number of defective parts in
the next 25 parts produced.
3. Each sample of air has a 10% chance of containing a particular rare molecule. Let X =
the number of air samples that contain the rare molecule in the next 18 samples analyzed.
4. Of all bits transmitted through a digital transmission channel, 10% are received in error.

Let X = the number of bits in error in the next ive bits transmitted.
5. A multiple-choice test contains 10 questions, each with four choices, and you guess at each
question. Let X = the number of questions answered correctly.
6. In the next 20 births at a hospital, let X = the number of female births.
7. Of all patients suffering a particular illness, 35% experience improvement from a particular
medication. In the next 100 patients administered the medication, let X = the number of
patients who experience improvement.
These examples illustrate that a general probability model that includes these experiments as
particular cases would be very useful.
Each of these random experiments can be thought of as consisting of a series of repeated, random

trials: 10 lips of the coin in experiment 1, the production of 25 parts in experiment 2, and so forth.
The random variable in each case is a count of the number of trials that meet a specii ed criterion.
The outcome from each trial either meets the criterion that X counts or it does not; consequently,
each trial can be summarized as resulting in either a success or a failure. For example, in the mul-
tiple-choice experiment, for each question, only the choice that is correct is considered a success.
Choosing any one of the three incorrect choices results in the trial being summarized as a failure.
The terms success and failure are just labels. We can just as well use A and B or 0 or 1.
Unfortunately, the usual labels can sometimes be misleading. In experiment 2, because X counts
defective parts, the production of a defective part is called a success.
A trial with only two possible outcomes is used so frequently as a building block of a ran-
dom experiment that it is called a Bernoulli trial. It is usually assumed that the trials that con-
stitute the random experiment are independent. This implies that the outcome from one trial
has no effect on the outcome to be obtained from any other trial. Furthermore, it is often rea-
sonable to assume that the probability of a success in each trial is constant. In the multiple-
choice experiment, if the test taker has no knowledge of the material and just guesses at each
question, we might assume that the probability of a correct answer is 1 4 for each question.

EXAMPLE 3-16 Digital Channel The chance that a bit transmitted through a digital transmission channel is
received in error is 0.1. Also, assume that the transmission trials are independent. Let X = the
(
number of bits in error in the next four bits transmitted. Determine P X = ) 2 .
Let the letter E denote a bit in error, and let the letter O denote that the bit is okay, that is, received without error.
We can represent the outcomes of this experiment as a list of four letters that indicate the bits that are in error and