WEBVTT
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We end this lecture sequence
with the most important
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property of expectations,
namely linearity.
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The idea is pretty simple.
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Suppose that our random
variable, X, is the salary of
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a random person out of
some population.
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So that we can think of the
expected value of X as the
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average salary within
that population.
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And now suppose that everyone
gets a raise, and
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Y is the new salary.
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And generously, the new salary
is twice the old salary plus a
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bonus of $100.
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What happens to the expected
value of the salary, or the
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average salary?
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Well the new average salary,
which is the expected value of
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2X plus 100, is twice the
old average plus 100.
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So doubling everyone's salary
and giving to everyone an
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additional $100, what it does
to the average is that it
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doubles the average and
adds 100 to it.
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This is the linearity property
of expectation in one
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particular example.
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It's a most intuitive property,
but it's worth also
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deriving it in a formal way.
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And the derivation proceeds
through the
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expected value rule.
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We're dealing here with a
particular function, g, which
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is a linear function.
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So we're dealing with a linear
function, ax plus b.
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And we're dealing with a random
variable, Y, which is g
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applied to an original
random variable, X.
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So the expected value of Y can
be calculated according to the
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expected value rule.
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It's the sum over all x's of g
of x times the probability of
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that particular x.
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And we plug-in the specific form
of the function, g, which
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is ax plus b.
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And then we separate the
sum into two sums.
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The first sum, after pulling
out a constant of
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a, takes this form.
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And the second sum, after
pulling out the constant, b,
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takes this form.
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Now, the first sum is a times
the expected value of X. This
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is just the definition of
the expected value.
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As, for the second sum, we
realize that this quantity is
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equal to 1 because it is the
sum of the probabilities of
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all the different values of X.
And this concludes the proof
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of the linearity of
expected values.
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Notice that for expected values,
what we have is that
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the expected value of Y, which
is expected value of g of X,
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is this same as g of the
expected value of X. The
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expected value of a linear
function is the same linear
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function applied to the
expected value.
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But this is an exceptional
case.
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This does not happen
in general.
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It's an exceptional function
g that makes this happen.
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This property is true for
linear functions.
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But for non-linear functions,
it is generally false.