CS 70
Discrete Mathematics and Probability Theory
Fall 2010
Tse/Wagner
Note 14
Random Variables: Distribution and Expectation
Random Variables
Question
: The homeworks of 20 students are collected in, randomly shuffled and returned to the students.
How many students receive their own homework?
To answer this question, we first need to specify the probability space: plainly, it should consist of all 20!
permutations of the homeworks, each with probability
1
20!
. [Note that this is the same as the probability
space for card shuffling, except that the number of items being shuffled is now 20 rather than 52.] It helps
to have a picture of a permutation. Think of 20 books lined up on a shelf, labeled from left to right with
1
,
2
,...,
20. A permutation
π
is just a reordering of the books, which we can describe just by listing their
labels from left to right. Let’s denote by
π
i
the label of the book that is in position
i
. We are interested in the
number of books that are still in their original position, i.e., in the number of
i
’s such that
π
i
=
i
. These are
often known as
fixed points
of the permutation.
Of course, our question does not have a simple numerical answer (such as 6), because the number depends on
the particular permutation we choose (i.e., on the sample point). Let’s call the number of fixed points
X
. To
make life simpler, let’s also shrink the class size down to 3 for a while. The following table gives a complete
listing of the sample space (of size 3!
=
6), together with the corresponding value of
X
for each sample
point. [We use our bookshelf convention for writing a permutation: thus, for example, the permutation 312
means that book 3 is on the left, book 1 in the center, and book 2 on the right. You should check you agree
with this table.]
permutation
π
value of
X
123
3
132
1
213
1
231
0
312
0
321
1
Thus we see that
X
takes on values 0, 1 or 3, depending on the sample point. A quantity like this, which
takes on some numerical value at each sample point, is called a
random variable
(or
r.v.
) on the sample
space.
Definition 14.1 (random variable)
:
A
random variable X
on a sample space
Ω
is a function that assigns
to each sample point
ω
∈
Ω
a real number
X
(
ω
)
.
Until further notice, we’ll restrict out attention to random variables that are
discrete
, i.e., they take values in
a range that is finite or countably infinite.
The r.v.
X
in our permutation example above is completely specified by its values at all sample points, as
CS 70, Fall 2010, Note 14
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Figure 1: Visualization of how a random variable is defined on the sample space.
given in the above table. (Thus, for example,
X
(
123
) =
3 etc.)
A random variable can be visualized in general by the picture in Figure 1
1
. Note that the term “random
variable” is really something of a misnomer: it is a function so there is nothing random about it and it is
definitely not a variable! What is random is which sample point of the experiment is realized and hence the
value that the random variable maps the sample point to.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Rau
 Probability theory

Click to edit the document details