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Section 8.1
Random Variables and Distributions
Variable: A characteristic that varies from one person or thing to
another.
Random variables: is simply a variable that takes on numerical
values that depend on the outcomes of a chance operation.
Numerical- numerically valued variables.
1. Discrete variable is a quantitative variable whose
possible values can be listed (also listed
indefinitely). Examples would include number of
siblings, number of students in a class, different
rolls of a die, and flip of a coin.
a. Some discrete variables have an infinite
number of outcomes such as the scores of a
football game.
b. Some discrete variables have a finite number
of outcomes such as the list of values for a roll
of a die.
2. Continuous variable is a quantitative variable
whose possible values form some sort of interval
of numbers such as height and time of birth.
Notation for Variables and Observations
X : denotes the variable itself
x : denotes the value or observation of the variable
Example 1
Classify each random variable X as finite, discrete infinite, or
continuous, and indicate the values that X can take.
5. Look at the second hand of your watch; X is the time it reads in
seconds.
6. Watch a soccer game, X =the total number of goals scored.
Frequency: is how often a value for a quantitative variable occurs.
Frequency Distribution: is a listing of the distinct values of
quantitative data, and how often they occur. A frequency
distribution can be displayed in a tabular form or in a graph.
Relative frequency: is the ratio of the frequency to the total number
of observations.
Relative frequency 
Frequency
Number of observations (n)
Relative frequency percentage 
Frequency
100%
Number of observations (n)
Procedure 1: Frequency Distribution for Quantitative Data
1. List the distinct values of the observations for the variable in
the data set in the first column of a table.
2. Record the frequency or relative frequency in the second
column of the table.
A relative frequency distribution is also known as a probability
distribution.
Example 2
12. X is the largest number of consecutive times heads comes up
in a row when a coin is tossed three times.
A. Show what the sample space is
B. Complete the following sentence: “ X is the rule that assigns to
each …….”
C. List the values of X for all of the outcomes.
18. The capacities of the hard drive of your dormitory suite mates’
computers are 1,000 GB, 1,500 GB, 2,000 GB, 2,500 GB, 3,000
GB, 3,500 GB.
A. Show what the sample space is
B. Complete the following sentence: “ X is the rule that assigns to
each …….”
C. List the values of X for all of the outcomes.
We can also use relative frequency distributions (probability
distributions) to answer probability questions
Example 3
20. The random variable X has the probability distribution table
shown below:
-2
-1
0
1
2
x
P( X  x)
0.4
0.1
0.1
a. Calculate P( X  0), P( X  0)
b. Assuming P( X  2)  P( X  1) , find each of the missing
values.
Procedure 2: Construct a Histogram
1. Obtain a frequency (relative frequency, percent) distribution of
the numerical data.
2. Draw a horizontal axis on which to place the bars and a vertical
axis to display the frequencies (relative frequencies,
percentage)
3. For each distinct value, construct a vertical bar whose height
equals the frequency (relative frequency, percent) of that class.
4. Label the bars with the distinct values, the horizontal axis with
the name of the variable, and the vertical axis with
“Frequency”( “Relative Frequency,” “Percent”).
Example 4
22. A fair die is rolled, and X is the square of the number facing up.
Calculate P( X  9)
Give the probability distribution for the indicated random variable
and draw the corresponding histogram and calculate the indicated
probability.
30. 2003 Income Distribution up to $100,000. Use the following
data from a sample of 1,000 households in the U.S. in 2003.
Income
0-19,999 20,00040,00060,00080,000Bracket ($)
39,999
59,999
79,999
99,999
Households 270
280
200
150
100
a. Let X be the (rounded) midpoint of a bracket in which a
household falls. Find the relative frequency distribution of X
and graph its histogram.
b. Shade the area of your histogram corresponding to the
probability that a random selected U.S. household in the
sample has a value of X above 50,000. What is this
probability?
34. The following table shows the average percentage increase in
the price of a house from 1980-2001 in nine regions of the United
States.
New England
300
Pacific
225
Middle Atlantic
225
South Atlantic
150
Mountain
150
West North Central
125
West South Central
75
East North Central
150
East South Central
125
Let X be the percentage increase in the price of a house in a
randomly selected region.
a. What are the values of X ?
b. Compute the frequency and probability distribution of X
c. What is the probability that, in a randomly selected region, the
percentage increase in the cost of a house exceeded 200%?