WEBVTT
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We now note some elementary
properties of expectations.
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These will be some properties
that are extremely natural and
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intuitive, but even so, they
are worth recording.
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The first property
is the following.
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If you have a random variable
which is non-negative, then
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its expected value is
also non-negative.
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What does it mean that the
random variable is
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non-negative?
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What it means is that for all
possible outcomes of the
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experiment, no matter what the
outcome is, the associated
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numerical value of the
random variable is a
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non-negative number.
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What's the implication
of this?
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When we calculate an expectation
we're adding over
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all the possible numerical
values of the random variable.
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All the possible numerical
values of the random variable
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under this assumption
are non-negative.
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Probabilities are also
non-negative.
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So we have a sum of non-negative
entries and
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therefore, the expected value
is also going to be
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non-negative.
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The next property is a
generalization of this.
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Consider now a random variable
that has the property that no
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matter what the outcome of the
experiment is, the value of
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this random variable lies
in the range between two
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constants, a and b.
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In this case, we argue
as follows.
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The expected value, by
definition, is a sum over all
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possible values of the random
variable of certain terms.
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Now, the possible numerical
values of the random variable
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are all of them at least as
large as a, so this gives us
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an inequality of this type.
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Then, we pull a factor of a
outside of the summation.
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And finally, we recall that
the sum of a PMF over all
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possible values of little
x is equal to 1.
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Why is that the case?
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Well, these are the
probabilities for the
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different numerical values
of the random variable.
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The sum of the probabilities of
all the possible numerical
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values has to be equal to 1,
because that exhausts all the
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possibilities.
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So we obtain a times
1, which is a.
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So, what we have proved is that
the expected value is at
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least large as a.
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You can use a symmetrical
argument where the
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inequalities will go the
opposite way and where a's
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will be replaced by b's,
to prove the second
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inequality, as well.
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The last fact we want to take
note of is the following.
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If we have a constant and we
take its expected value, we
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obtain the same constant.
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What does that mean?
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We have only been talking
about expected values of
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random variables.
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What does it mean to take the
expected value of a constant?
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Well, as we discussed earlier,
we can think of a constant as
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being a random variable of
a very special type.
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A random variable whose
PMF takes this form.
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This random variable can take
only a single value and the
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probability of that single
value is equal to 1.
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This means that in the formula
for the expected value there's
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going to be only one term in
this summation, and that term
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is going to be c times the
probability that our random
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variable takes the value c.
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Now, that probability is equal
to 1, and we're left with c.
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So this equality makes sense,
of course, as long as you
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understand that a constant can
also be viewed as a random
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variable of a very
degenerate type.
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Now, intuitively, of course,
it's certainly clear
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what this is saying.
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That if a certain quantity is
always equal to c, then on the
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average, it will also
be equal to c.