After this section, you will be able to:
1.
Distinguish between discrete and continuous random variables.
2.
Graph discrete probability distributions.
3.
Compute 𝜇 and 𝜎 for a discrete probability distribution.
4.
Compute 𝜇 and 𝜎 for a linear function of a random variable x.
5.
Compute 𝜇 and 𝜎 for a linear combination of two independent random variables.
______________: a variable (typically x) that has a single _________________,
determined by ________________, for each outcome of a procedure.
_____________ random variable: can take on only a ___________ number of values
or a ___________________ number of values. “________________” refers to the fact
that there might be ____________________ many values, but they result from the
counting process.
_____________ random variable: can take on any of the countless number of values in
a __________________________________. There are no gaps or interruptions.
Which of the following random variables are discrete and which are continuous?
1. Let x = the number of eggs that a hen lays in a day.
2. The count of the number of patrons attending a rock concert.
3. Let x = the amount of milk a cow produces in a day.
4. The measure of voltage for a smoke-detector battery.
x (girls)
P(x)
0
0.000
1
0.001
2
0.006
_________________, or ___________________ that gives the
3
0.022
probability for each value of the random variable.
4
0.061
5
0.122
6
0.183
7
0.209
8
0.183
9
0.122
10
0.061
11
0.022
12
0.006
13
0.001
14
0.000
A probability distribution is a ________________,
Features:
1. The probability distribution has a probability assigned to
______________ distinct value of the random variable.
2. The sum of all the assigned probabilities must be ______.
0.25
Probability
0.2
Probability
Histogram
0.15
0.1
0.05
0
1
2
3
4
5
6
7
8
9
10
11
12
Number of Girls among Fourteen Newborns
13
14
Dr. Mendoza developed a test to measure boredom tolerance. He
administered it to a group of 20,000 adults between the ages of 25 and
35. The possible scores were 0, 1, 2, 3, 4, 5, and 6, with 6 indicating
the highest tolerance for boredom. Here are the results:
Boredom Tolerance Test Scores
20,000 subjects
Score
Number of Subjects
0
1400
1
2600
2
3600
3
6000
4
4400
5
1600
6
400
Probability Distribution of
Scores on Boredom Tolerance Test
Score x
0
1
2
3
4
5
6
Probability P(x)
continued
b. Draw a relative-frequency
histogram for the test scores.
c. The Topnotch Clothing
Company needs to hire someone
with a score on the boredom
tolerance test of 5 or 6 to
operate the fabric press
machine. What is the probability
that someone will be randomly
chosen from the group who took
the test and scored either 5 or 6?
Referring to the probability distribution of a __________________, we use:
_________ for the mean
________ =
𝑥𝑃(𝑥)
_______ for the standard deviation
________ =
(𝑥 − 𝜇)2 𝑃(𝑥)
• X is the value of the random variable
• P(x) is the probability of that variable
• The sum
is taken for all values of the random variable.
Are we influenced to buy a product by an ad we saw on TV? National
Infomercial Marketing Association determined the number of times buyers of
a product watched a TV infomercial before purchasing the product. The
results are as follows:
Number of Times Buyers Saw Infomercial
Percentage of Buyers
1
2
3
4
5
27%
31%
18%
9%
15%
We can treat the information shown as an estimate of the probability
distribution because he events are mutually exclusive and the sum of the
percentages is 100%.
Compute the mean and standard deviation of the distribution.
continued
Number of Times Buyers Saw Infomercial
Percentage of Buyers
𝜇=
𝑥𝑃(𝑥)
1
2
3
4
5
27%
31%
18%
9%
15%
𝜎=
(𝑥 − 𝜇)2 𝑃(𝑥)
Number of Times Buyers View Infomercial Before Making Purchase
x
P(x)
1
2
3
0.27
0.31
0.18
4
5
0.09
0.15
Page 178-179
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