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Section 7.1
Understanding Random Variables
What is a RANDOM variable?
By the BOOK:
A random variable is a variable whose
value is a numerical outcome of a
random phenomenon.
You will be expected to DEFINE a
random variable by clearly describing it
based on the given information.
Example Scenario
You are taking a five question True/False quiz
in your AP Statistics course. You are really
concerned with how many questions you are
able to get correct.
Define the random variable, X, that is present
in this situation.
The random variable X is defined as the number of
questions answered correctly.
DISCRETE Random Variable
By the BOOK:
A discrete random variable X has a
countable number of possible values.
The probability distribution for a discrete random
variable will look very similar to the probability
models from last chapter.
Discrete Random Variable
Example
Recently a friend of yours created a game.
He tells you that you are going to get three
cards in exchange for two dollars. Your
winnings will be based on the number of
spades you get. You will get a dollar for
each spade. Define the random variable X
and describe the probability distribution.
Example Solution
X = the number of dollars profited
X
-2
-1
0
1
P(X)
.4135
.4359
.1376
.0129
CONTINUOUS Random Variable
By the BOOK:
A continuous random variable X takes all
values in an interval of numbers.
The probability distribution of a continuous
random variable will be described by a density
curve. Probabilities will be calculated based on
area underneath the curve.
Any density curve (uniform, normal, etc.) can be
used to describe a continuous random variable.
Continuous Random Variable
Example
In high school there is always focus on
GPA. At Walton High School students tend
to do better than they do nationally. It has
been established that Walton GPAs are
normally distributed with a mean of 3.1 and
a standard deviation of 0.47. Define the
random variable, X, and describe the
probability distribution.
Example Solution
X = the amount of grade points
obtained per class on average
The probability distribution of X can
be defined by N(3.1, 0.47).
Other Important Information
Recall, the probability that a continuous
random variable takes one specific value
will be zero (0) because there is no area
under the curve for one point.
To get a clear picture of a probability
distribution you can look at a probability
histogram (for discrete) or the density curve
(for continuous).
Defining Random Variables
 As a general guideline, when you are
defining a DISCRETE random variable,
start with the phrase “the number of . . .”
 When you are defining a CONTINUOUS
random variable, start with the phrase “the
amount of . . .” This is often harder to
verbalize, but by trying to use this phrase it
will hopefully help clarify what your
variable represents
Review Problems
A. Assume that the lengths of salmon are normally
distributed with a mean of 10.5 inches and a
standard deviation of 2.3 inches.
1. What is the probability of catching a salmon that is
more than a foot long? At least a foot long?
P(X >12)=P(X≥12)=normalcdf(12,E99,10.5,2.3)=0.2571
B. Consider choosing a number,X, between 0 and 5
where the distribution is uniform.
2. P(X=2) = 0
3. P(X>2.3) = 0.54
4. P(X<1.4 or X>4.8) = 0.32