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Random Variables
Probability Continued
Chapter 7
Random Variables
Suppose that each of three randomly selected
customers purchasing a hot tub at a certain
store chooses either an electric (E) or a gas (G)
model. Assume that these customers make
their choices independently of one another and
that 40% of all customers select an electric
model. The number among the three
customers who purchase an electric hot tub is a
random variable. What is the probability
distribution?
Random Variable Example
X = number of people who purchase electric hot tub
X 0
1
2
3
P(X) .216 .432 .288 .064
GGG
(.6)(.6)(.6)
EGG
GEG
GGE
(.4)(.6)(.6)
(.6)(.4)(.6)
(.6)(.6)(.4)
EEG
GEE
EGE
(.4)(.4)(.6)
(.6)(.4)(.4)
(.4)(.6)(.4)
EEE
(.4)(.4)(.4)
Random Variables
A numerical variable whose value
depends on the outcome of a chance
experiment is called a random
variable.
discrete versus continuous
Discrete vs. Continuous
The number of desks in a classroom.
The fuel efficiency (mpg) of an
automobile.
The distance that a person throws a
baseball.
The number of questions asked during a
statistics final exam.
Discrete versus Continuous
Probability Distributions
Which is which?
Properties:
For every possible x value, 0 < x < 1.
 Sum of all possible probabilities add to 1.

Properties:
Often represented by a graph or function.
 Area of domain is 1.

Probability Histograms
We can create a probability histogram
to show the distributions of discrete
random variables.
Example
Let X represent the sum of two dice.
Then the probability distribution of X is
as follows:
X
2
3
4
5
6
7
8
9
10
11
12
P(X)
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
Continuous Random Variable
and Density Curves
The probability distribution of a
continuous random variable assigns
probabilities under a density curve.
Probabilities are assigned to
INTERVALS of outcomes rather than to
individual outcomes.
A probability of 0 is assigned to every
individual outcome in a continuous
probability distribution.
The Normal Distribution can
be a Probability Distribution
The normal curve
Means and Variances
The mean value of a random variable X
(written mx ) describes where the probability
distribution of X is centered.
We often find the mean is not a possible
value of X, so it can also be referred to as
the “expected value.”
The standard deviation of a random
variable X (written sx )describes variability
in the probability distribution.
Mean of a Random Variable
Example
Below is a distribution for number of
visits to a dentist in one year. X = # of
visits to the dentist.
X
0 1 2 3
4
P( X ) .1 .3 .4 .15 .05
Determine the expected value,
variance and standard deviation.
Formulas
Mean of a Random Variable
m X   xi pi
Variance of a Random Variable
s X2   ( xi  m X ) 2 pi
Mean of a Random Variable
Example
X
0 1 2 3
4
P( X ) .1 .3 .4 .15 .05
m X   xi pi
E(X) = 0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05)
= 1.75 visits to the dentist
Variance and Standard Deviation of
a Random Variable Example
X
0 1 2 3
4
P( X ) .1 .3 .4 .15 .05
s   ( xi  m X ) pi
Var(X) = (0 – 1.75)2(.1) + (1 – 1.75)2(.3) +
(2 – 1.75)2(.4) + (3 – 1.75)2(.15) +
= .9875
(4 – 1.75)2(.05)
2
X
2
s X  .9875  .9937 visits