Random Variables
Lesson 5.2
Analyzing Discrete Random Variables
Statistics and Probability with Applications, 3rd Edition
Starnes & Tabor
Bedford Freeman Worth Publishers
Analyzing Discrete Random Variables
Learning Targets
After this lesson, you should be able to:
 Make a histogram to display the probability distribution of a
discrete random variable and describe its shape.
 Calculate and interpret the mean (expected value) of a discrete
random variable.
 Calculate and interpret the standard deviation of a discrete
random variable.
Statistics and Probability with Applications, 3rd Edition
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Analyzing Discrete Random Variables
Consider the discrete random variable X = Apgar score of a randomly
selected baby one minute after birth.
We can display the probability distribution in a histogram. Values of the
variable go on the horizontal axis and probabilities go on the vertical
axis. There is one bar in the histogram for each value of X. The height
of each bar gives the probability for the corresponding value of the
variable.
Statistics and Probability with Applications, 3rd Edition
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Analyzing Discrete Random Variables
The mean (expected value) of any discrete random variable is an
average of the possible outcomes, but a weighted average in which
each outcome is weighted by its probability.
Mean (Expected Value) of a Discrete Random Variable
The mean (expected value) of a discrete random variable is its
long-run average value over many, many repetitions of the same
chance process.
Suppose that X is a discrete random variable with probability
distribution
To find the mean (expected value) of X, multiply each possible value
of X by its probability, then add all the products:
Statistics and Probability with Applications, 3rd Edition
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Analyzing Discrete Random Variables
With the mean as our measure of center for a discrete random variable,
it shouldn’t surprise you that we’ll use the standard deviation as our
measure of spread.
Standard Deviation of a Discrete Random Variable
The standard deviation of a discrete random variable measures how
much the values of the variable typically differ from the mean.
Suppose that X is a discrete random variable with probability
distribution
and that µX is the mean of X. The standard deviation of X is
Statistics and Probability with Applications, 3rd Edition
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LESSON APP 5.2
How much do college grades vary?
Indiana University Bloomington posts the grade distributions for its
courses online. In a recent semester of a Business Statistics course,
45.7% of students received A’s, 36.2% B’s, 13.8% C’s, 3.2% D’s, and
1.1% F’s. Choose a Business Statistics student at random.
The student’s grade on a 4-point scale (with A = 4) is a random variable
X with this probability distribution:
1. Make a histogram of the probability distribution. Describe its shape.
2. Calculate and interpret the mean of X.
3. Calculate and interpret the standard deviation of X.
Statistics and Probability with Applications, 3rd Edition
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Analyzing Discrete Random Variables
Learning Targets
After this lesson, you should be able to:
 Make a histogram to display the probability distribution of a
discrete random variable and describe its shape.
 Calculate and interpret the mean (expected value) of a discrete
random variable.
 Calculate and interpret the standard deviation of a discrete
random variable.
Statistics and Probability with Applications, 3rd Edition
7