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When we have independence,
does anything interesting

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happen to expectations?

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We know that, in general, the
expected value of a function

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of random variables is not the
same as applying the function

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to the expected values.

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And we also know that there are
some exceptions where we

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do get equality.

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This is the case where we are
dealing with linear functions

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of one or more random
variables.

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Note that this last property
is always true and does not

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require any independence
assumptions.

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When we have independence,
there is one additional

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property that turns
out to be true.

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The expected value of the
product of two independent

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random variables is
the product of

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their expected values.

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Let us verify this relation.

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We are dealing here with the
expected value of a function

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of random variables, where the
function is defined to be the

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product function.

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So to calculate this expected
value, you can use the

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expected value rule.

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And we are going to get the sum
over all x, the sum over

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all y, of g of xy, but in this
case, g of xy is x times y.

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And then we weigh all those
values according to the

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probabilities as given
by the joint PMF.

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Now, using independence, this
sum can be changed into the

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following form--

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the joint PMF is the product
of the marginal PMFs.

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And now when we look at the
inner sum over all values of

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y, we can take outside the
summation those terms that do

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not depend on y, and so this
term and that term.

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And this is going to yield a
summation over x of x times

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the marginal PMF of X, and then
the summation over all y

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of y times the marginal PMF of
Y. But now we recognize that

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here we have just the expected
value of Y. And then we will

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be left with another expression,
which is the

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expected value of X. And this
completes the argument.

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Now, consider a function of X
and another function of Y. X

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and Y are independent.

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Intuitively, the value of X
does not give you any new

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information about Y, so the
value of g of X does not to

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give you any new information
about h of Y. So on the basis

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of this intuitive argument, the
functions g of X and h of

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Y are also independent
of each other.

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Therefore, we can apply the
fact that we have already

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proved, but with g of X in the
place of X and h of Y in the

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place of Y. And this gives us
this more general fact that

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the expected value of the
product of two functions of

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independent random variables is
equal to the product of the

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expectations of these
functions.

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We could also prove this
property directly without

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relying on the intuitive
argument.

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We could just follow the same
steps as in this derivation.

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Wherever there is an X, we
would write g of X, and

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wherever there is a Y, we would
write h of Y. And the

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same algebra would go through,
and we would end up with the

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expected value of g of X times
the expected value of h of Y.