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We are defining a random
variable as a real valued

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function on the sample space.

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So this is a good occasion to
make sure that we understand

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what a function is.

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To define a function, we
start with two sets.

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One set--

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call it A--

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is the domain of the function.

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And we have our second set.

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Then a function is a rule that
for any element of A

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associates an element of B. And
we use a notation of this

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kind to indicate that we are
dealing with a function f that

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maps elements of A into
elements of B.

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Now, two elements of A may be
mapped to the same element of

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B. This is allowed.

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What is important, however, is
that every element of A is

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mapped to exactly one element
of B, not more.

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But it is also possible that
we have some elements of B

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that do not correspond to any
of the elements of A.

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Now, I said that a function is
a rule that assigns points of

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A to points in B. But what
exactly do we mean by a rule?

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If we want to be more precise,
a function would

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be defined as follows.

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It would be defined as a
set of pairs of values.

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It would be a set of pairs of
the form x, y such that x is

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always an element of A, y is
always an element of B, and

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also-- most important--

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each x in A appears in
exactly one pair.

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So this would be a formal
definition of

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what a function is.

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It is collection of ordered
pairs of this kind.

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As a concrete example, let us
start with the set consisting

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of these elements here.

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And let B be the set
of real numbers.

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And consider the function that
corresponds to what we usually

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call the square.

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So it's a function that
squares its argument.

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Then this function would be
represented by the following

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collection of pairs.

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So this is the value of x.

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And this is the corresponding
value of y.

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Any particular x shows
up just once in this

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collection of pairs.

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But a certain y--

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for example, y equal to 1--

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shows up twice, because minus
1 and plus 1 both map to the

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same element of B.

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Now, this is a representation
in terms of ordered pairs.

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But we could also think of
the function as being

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described by a table.

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We could, for instance, put
this information here in a

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form of a table of this kind
and say that this table

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describes the function.

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For any element x, it
tells us what the

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corresponding element y is.

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However, when the set A is an
infinite set it is not clear

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what we might mean by saying
a table, an infinite table,

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whereas this definition
in terms of

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ordered pairs still applies.

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For example, if you're
interested in the function

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which is, again, the square
function from the real

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numbers, the way you would
specify that function

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abstractly would
be as follows.

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You could write, it's the set of
all pairs of this form such

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that x is a real number.

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And now such pairs, of course,
belong to the two dimensional

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plane because it's a
pair of numbers.

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So this set here can be viewed
as a formal definition or a

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specification of the
squaring function.

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Now, what this set is
is something that we

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can actually plot.

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If we go in the two dimensional
plane, the points

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of this form are exactly the
points that belong to the

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graph of the square function.

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So this abstract definition,
really all that it says is

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that a function is the
same thing as the

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plot of that function.

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But it's important here
to make a distinction.

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The function is the
entire plot--

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so this set here is
the function f--

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whereas if I tell you a
specific number x, the

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corresponding value here
would be f of x.

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So here x is a number and
f of x is also a number.

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And those two values, x and f
of x, define this particular

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point on this plot.

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But the function itself
is the entire plot.

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Let us also take this occasion
to talk a little bit about the

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notation and the proper way of
talking about functions.

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Here is the most common
way that one

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would describe a function.

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And this is an appropriate
way.

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We've described the domain.

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We've described the
set on which the

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function takes values.

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And I'm telling you for any x in
that set what the value of

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the function is.

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On the other hand, sometimes
people use a more loose

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language, such as for example,
they would say,

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the function x squared.

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What does that mean?

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Well, what this means is
exactly this statement.

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Now let us consider
this function.

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The function f--

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again, from the reals
to the reals--

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that's defined by f of
z equal to z squared.

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Is this a different function
or is it the same function?

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It's actually the same function,
because these two

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involve the same sets.

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And they produce their outputs,
the values of f,

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using exactly the same rule.

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They take an argument and they
square that argument.

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Now, if you were to use informal
notation, you would

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be referring to that
second function as

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the function z squared.

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And now, if you use informal
language, it's less clear that

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the function x squared and the
function z squared are one and

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the same thing, whereas with
this terminology here, it

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would be pretty clear
that we're talking

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about the same function.

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Finally, suppose that we have
already defined a function.

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How should we refer
to it in general?

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Should we call it the function
f, or should we say the

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function f of x?

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Well, when x is a number,
f of x is also a number.

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So f of x is not really
a function.

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The appropriate language
is this one.

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We talk about the function f,
although quite often, people

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will abuse language and they
will use this terminology.

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But it's important to keep in
mind what we really mean.

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The idea is that we need to
think of a function as some

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kind of box or even a computer
program, if you wish, that

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takes inputs and produces
outputs.

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And there's a distinction
between f, which is the box,

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from the value f of x that the
function takes if we feed it

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with a specific argument.