WEBVTT
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We now start with our agenda
of developing continuous
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counterparts of everything
we have done for
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discrete random variables.
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Let us look at the concept
of expectation.
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In the discrete case, we have
defined expectation as a
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weighted average of the values
X of the random variable,
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weighted according to their
corresponding probabilities.
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In the continuous case, we
define expectation in a
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similar way--
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as a weighted average over the
possible values of X, weighted
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according to the corresponding
value of the density.
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Points where the density
is higher--
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for example, here--
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will receive a higher weight
in this calculation.
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But of course, since we are
averaging over a continuous
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set, the summation will have to
be replaced by an integral.
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This will be a recurrent
theme in this unit.
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Definitions or formulas for
the continuous case look
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exactly like the discrete ones,
except that PMFs are
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replaced by densities,
as here.
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The PMF is replaced
by a density.
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And summations are replaced
by integrals.
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The intuition is usually the
same in both the discrete and
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the continuous case.
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However, the intuition is
usually much clearer, much
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easier to visualize in
the discrete case.
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So the best strategy is to make
sure to understand fully
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the intuition for the discrete
case and just rely on it.
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At this point, let me add
some fine print--
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a mathematical side point.
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This integral or the expectation
will not be always
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well defined.
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For this integral to make sense,
we will need to make
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the assumption that the integral
of the absolute value
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of little x, weighted according
to the density,
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gives us a finite result.
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Unless we explicitly say
something different, we will
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always assume that we're dealing
with random variables
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that satisfy this condition.
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And so the expectation is well
defined mathematically.
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Coming back to the big
picture, regarding
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expectations, the intuition
remains the same as in the
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discrete case-- that the
expectation represents the
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average of the values we expect
to see in a very large
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number of independent
repetitions of the experiment.
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In fact, there are also theorems
to this effect, but
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these will have to wait until
later in this class when we
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study limit theorems.
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Another intuitive interpretation
that is true
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for both the discrete and the
continuous case is that the
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expectation corresponds to the
center of gravity of the
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probability distribution.
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So in this diagram, it might
be somewhere around here.
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And similarly, for the
continuous diagram, the center
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of gravity might be somewhere
around here.
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And if it happens that the
distribution, the PMF or the
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PDF, happens to be symmetric
around a certain point, then
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that point will be equal
to the expectation.
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Expectations of continuous
random variables have all the
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properties you might expect.
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For example, non-negative
random variables have
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non-negative expectations.
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Random variables that lie
inside an interval have
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average values or expectations
that also lie
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inside the same interval.
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The derivation is exactly the
same as for the discrete case.
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There is also an expected
value rule.
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In the discrete case, it
took on this form.
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In the continuous case, we
obtain an analogous form in
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which the summation is replaced
by [an] integral.
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And instead of weighing
according to the PMF, we now
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weigh according to the
density function.
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The derivation of the expected
value rule for the continuous
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case is a little more
complicated than the one that
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we gave for the discrete case.
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But it's sufficient for us to
know that it is true and that
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it has an intuitive meaning that
runs along the same lines
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as the intuitive meaning that we
had for the discrete case.
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As an instance of how we might
apply the expected value rule,
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if you wish to calculate the
expected value of the square
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of a continuous random
variable, you
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would proceed as follows.
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You would integrate over the
entire real line the value of
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the function, which is X squared
in our case, weighted
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according to the density.
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Finally, a most important
property of
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expectations, is linearity.
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Linearity is still true
for continuous random
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variables as well.
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And the way it is derived is
exactly the same as in the
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discrete case.
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Namely, we apply the expected
value rule to this function of
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the random variable
X and separate
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out the various terms.
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The story regarding variances is
exactly the same as in the
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discrete case.
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We define variances using
the same definition.
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And of course, here, mu stands
for the expected value of the
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random variable X.
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To calculate the variance, we
can use the expected value
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rule, which takes this form
in the continuous case.
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And we apply the expected value
rule for the case where
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we're dealing with the expected
value of this
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particular function, so that
in this instance, the
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functions g of x is x
minus mu squared.
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So by applying the expected
value rule, we obtain the
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integral from minus infinity to
infinity, the functions g
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of x, weighted according to the
density, and then we carry
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out the integration.
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We also define the standard
deviation--
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same way as in the
discrete case.
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We have a property about a
variance of linear functions,
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of a random variable, namely,
that if we add a constant to a
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random variable, this has no
effect on the variance.
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But if we multiply a random
variable by a constant, the
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variance gets multiplied by the
square of that constant.
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Finally, when calculating the
variance, it is often
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convenient to use this
alternative formula in which
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the variance is calculated by
finding the expected value of
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the square of the random
variable and also using the
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expected value of the random
variable, but squared and
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subtracted from the
first term.
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This relation and this relation
are both derived
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exactly the same way as
in the discrete case.
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And there's no reason to repeat
those derivations.