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Chapter 16
Random Variables
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Expected Value: Center

A random variable assumes a value based on the
___________ of a random event.
 We use a capital letter, like X, to denote a
__________ variable.
 A ______________ value of a random variable
will be denoted with a lower case letter, in this
case x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 2
Expected Value: Center (cont.)

There are two types of random variables:
 ____________ random variables can take one
of a finite number of distinct outcomes.


Example:
_____________ random variables can take
any numeric value within a range of values.

Example:
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Slide 16- 3
Expected Value: Center (cont.)


A _______________________ for a random
variable consists of:
 The collection of all ____________ values of a
random variable, and
 the ______________ that the values occur.
Of particular interest is the value we expect a
random variable to take on, notated μ (for
population mean) or E(X) for _______________.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 4
Expected Value: Center (cont.)

The expected value of a (discrete) random
variable can be found by summing the products
of each _______________ by the ____________
that it occurs:

Note: Be sure that ________ possible outcome is
included in the sum and verify that you have a
________ probability model to start with.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 5
First Center, Now Spread…

For data, we calculated the standard deviation by
first computing the deviation from the mean and
squaring it. We do that with random variables as
well.
The variance for a random variable is:

The standard deviation for a random variable is:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 6
To calculate the mean and stadard
deviation using the TI



Enter random variable values in ___ and enter the
associated probabilities in ___.
Calculate _________________.
Note that
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 7
Just Checking

For a particular mechanical problem, 75% of the time the problem can
be fixed by a simple $60 repair job. However, the rest of the time the
complex job will cost an additional $140.
A) Define the random variable and construct the probability model.

B)
What is the expected value of the cost of this repair?

C)
What does that mean in this context?

D)
What is the standard deviation of cost?

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 8
Assignment

P. 381 #1-8, 15-20
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 9
Chapter 16
Random Variables
(2)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
More About Means and Variances

Adding or subtracting a constant from data shifts
the _______ but doesn’t change the __________
or _____________________:
E(X ± c) = ________
Var(X ± c) = _______

Example: Consider everyone in a company
receiving a $5000 increase in salary.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 11
More About Means and Variances (cont.)

In general, multiplying each value of a random
variable by a constant multiplies the _______ by
that constant and the _________ by the _______
of the constant:
E(aX) = ______ Var(aX) = _________

Example: Consider everyone in a company
receiving a 10% increase in salary.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 12
More About Means and Variances (cont.)

In general,
 The mean of the sum of two random variables
is the ___________________.
 The mean of the difference of two random
variables is the _______________________.
E(X ± Y) = _____________

If the random variables are independent, the
variance of their sum or difference is always
the ____________________.
Var(X ± Y) = ___________________
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 13
Continuous Random Variables



Random variables that can take on any value in a
range of values are called ___________ random
variables.
Continuous random variables have means
(__________values) and variances.
We won’t worry about how to calculate these
means and variances in this course, but we can
still work with models for continuous random
variables when we’re given the _____________.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 14
Continuous Random Variables (cont.)



Good news: nearly everything we’ve said about
how discrete random variables behave is true of
continuous random variables, as well.
When two independent continuous random
variables have ________________, so does their
sum or difference.
This fact will let us apply our knowledge of
Normal probabilities to questions about the _____
or __________ of independent random variables.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 15
What Can Go Wrong?


Probability models are still just models.
 Models can be _________, but they are not
_________.
 Question probabilities as you would data, and
think about the ________________ behind
your models.
If the ________ is _______, so is everything else.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 16
What Can Go Wrong? (cont.)


Don’t assume everything’s _________.
Watch out for variables that aren’t ___________:
 You can add expected values for _____ two
random variables, but
 you can only add variances of _____________
random variables.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 17
What Can Go Wrong? (cont.)



Don’t forget: Variances of independent random
variables ____. Standard deviations _______.
Don’t forget: Variances of independent random
variables ____, even when you’re looking at the
_____________ between them.
Don’t forget: Don’t write independent instances of
a random variable with notation that looks like
they are the ______ variables.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 18
What have we learned?



We know how to work with random variables.
 We can use a probability model for a _______
random variable to find its ________________
and ______________________.
The mean of the sum or difference of two random
variables, discrete or continuous, is just the ____
_________________ of their means.
And, for independent random variables, the
variance of their sum or difference is always the
_____ of their variances.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 19
What have we learned? (cont.)

Normal models are once again special.
 Sums or differences of Normally distributed
random variables also follow _____________.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 20
Practice
The American Veterinary Association claims that the annual cost of
medical care for dogs averages $100, with a standard deviation of
$30, and for cats averages $120, with a standard deviation of $35.
a)
What’s the expected difference in the cost of medical care for dogs
and cats?
b)
c)
d)
What’s the standard deviation of that difference?
If the difference in costs can be described by a Normal model,
what’s the probability that medical expenses are higher for
someone’s dog than for her cat?
If a person has two dogs and a cat, what’s the probability that their
total annual expenses will be greater than $400?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 21
Assignment

P. 382 #21-23, 25, 28, 32, 33, 35, 38, 39
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 16- 22