WEBVTT
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We will now go through another
example to consolidate our
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intuition about the content
of the law of iterated
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expectations and the law
of the total variance.
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The example is as follows.
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We have a class, and that class
consists of 30 students
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in total who are divided
into sections--
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the first and the
second section.
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Let xi be the score of students
i, let's say the
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final grade in the class.
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We consider the following
probabilistic experiment.
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We pick a student at random,
uniformly, so that each
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student is equally likely
to be picked.
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And we define two random
variables--
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X is a numerical random variable
that gives us the
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score of the selected student.
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So if student i is selected,
the value of the random
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variable capital X is xi.
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And capital Y is defined as the
random variable, which is
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the section of the selected
student, so that y takes
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values 1 or 2.
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We're given some information.
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For the first section,
the average of the
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student scores is 90.
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For the second section,
the average of the
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student scores is 60.
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Given that information, what is
the expected value of the
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student score?
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Well, each student is equally
likely to be picked, so has
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probability 1 over
30 to be picked.
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And this multiplies the score of
the student, so this is the
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expected value of the random
variable of interest.
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What is this number?
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Well, we need to calculate
the sum of the xi's.
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The sum of the first 10 xi's is
equal to 90 times 10, and
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the sum of the xi's in
the other section is
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equal to 60 times 20.
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And we carry out the
calculation, and we find that
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the answer is 70.
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Now let us look at conditional
expectations.
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If Y is equal to 1, this means
that a student from section
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one was picked.
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And within that section, each
student is equally likely to
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be picked, so the outcome of
this random variable is
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equally likely to be any
one of these xi's.
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Each xi gets picked with
probability of 1 over 10.
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And so, the expected value of
this random variable is 90.
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Similarly for the second
section, the expected value of
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the score of a randomly selected
student, given that
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the student belongs in that
section, is equal to 60.
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With this information available,
now we can describe
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the abstract conditional
expectation,
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which is a random variable.
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This random variable takes the
value of 90 if a student from
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the first section was picked,
and the value of 60 if a
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student from the second
section was picked.
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What is the probability of this
event that the student
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from the first section
was picked?
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Given that the first section
has 10 out of a total of 30
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students, this probability is
1/3, and therefore, this
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probability is 2/3.
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Now that we have the
distribution of this random
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variable, we can calculate the
expected value of this random
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variable, which is 1/3 times
90 plus 2/3 times 60.
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And this number evaluates to 70,
which of course, it's no
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coincidence, it's the
same as the average
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over the entire class.
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By the law of iterated
expectations, we know that
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this quantity should be the
same as this quantity.
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So the law of iterated
expectations allows us to
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calculate the overall average in
the entire class by taking
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the section averages, and weigh
them according to the
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sizes of the different
sections.
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It's a divide and conquer
method, and it is similar to
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what we have been doing when we
use the total expectation
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theorem to divide and conquer.
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We continue with our example,
and here is a summary of what
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we found so far.
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The conditional expectation is
a random variable that takes
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these two values with certain
probabilities.
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And the mean of this random
variable is equal to 70.
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Let us now calculate
the variance
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of this random variable.
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This random variable, with
probability 1/3, takes a value
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90, which is this much away from
the mean of this random
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variable, which we square.
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And with probability 2/3, it
takes a value of 60, which is
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this much away from the mean
of the random variable.
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We square this, as well.
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And when we carry out the
calculation, we find that this
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number is equal to 200.
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Let us now continue.
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And suppose that somebody gave
us this piece of information.
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For the first section, this is
the deviation of the i-th
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student from the mean
of that section.
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So this is the sum of the
squares of the deviations and
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then we average over
all the students.
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We will use this data to
calculate certain quantities--
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for example, the variance
of the scores
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in the first section.
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Now in the first section, with
probability 1/10, we pick the
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ith student that
has this score.
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And this is the deviation of
that student from the mean of
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that section.
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So this is the same as the mean
squared deviation from
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the mean of the section.
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And this is exactly the variance
within that section.
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It is the variance of the random
variable, which is the
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score of a random student, given
that we are selecting a
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student from the
first section.
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For the second section,
the story similar.
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We're given this information,
and this tells us the variance
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of the student scores within
the second section.
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So now we can describe the
abstract conditional variance.
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It is a random variable that
takes this value with
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probability equal to the
probability of selecting
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someone from this section,
which is 1/3.
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Or it takes a value of 20, which
is the variance in the
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second section.
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And the second section is
selected with probability 2/3.
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With this information at hand,
now we can calculate the
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expected value of this random
variable, which is 1/3 times
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10 plus 2/3 times 20,
which is 50/3.
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At this point, we have the two
quantities that are necessary
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to apply the law of
total variance.
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According to the law of total
variance, the variance of the
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student scores throughout the
entire class is equal to this
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number, which is 50/3, plus
this number, which is 200.
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And this is the overall
variance.
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Now let us interpret
the law of total
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variance in this context.
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The interpretation
is as follows.
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The variance of the student
scores in the entire class
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consists of two pieces.
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The first piece looks at the
variance inside each section,
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which is 10 or 20, depending
on which section
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we're looking at.
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And we take the average over
the different sections.
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So we look at the variability of
the scores within a typical
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section, and then we average
over all the sections.
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The other term looks at the
means in the different
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sections, and figures out how
different are these means.
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How much do they vary from the
overall class average?
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It measures the variability
between different sections.
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So the overall randomness in the
test scores can be broken
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down into two pieces
of randomness.
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One source of randomness is that
the different sections
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have different means.
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The other source of randomness
is that inside each section,
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the students are different
from the
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means of their section.
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And these two pieces of
randomness together add up to
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the total randomness of the
student scores as measured by
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the variance of the
entire class.