WEBVTT
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We now move to a new topic--
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conditioning.
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Every probabilistic concept or
probabilistic fact has a
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conditional counterpart.
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As we have seen before, we can
start with a probabilistic
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model and some initial
probabilities.
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But then if we are told that
the certain event has
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occurred, we can revise our
model and consider conditional
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probabilities that take into
account the available
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information.
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But as a consequence, the
probabilities associated with
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any given random variable will
also have to be revised.
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So a PMF will have to be changed
to a conditional PMF.
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Let us see what is involved.
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Consider a random variable X
with some given PMF, whose
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values, of course, sum
to 1, as must be true
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for any valid PMF.
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We are then told that a certain
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event, A, has occurred.
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In that case, the event
that X is equal to--
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little x--
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will now have a conditional
probability of this form.
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We will use this notation here
to denote the conditional
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probability that the random
variable takes the
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value little x.
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Notice that the subscripts are
used to indicate what we're
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talking about.
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In this case, we are talking
about the random variable X in
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a model where event A is
known to have occurred.
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Of course, for this conditional
probability to be
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well defined, we will have to
assume that the probability of
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A is positive.
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This conditional PMF is like an
ordinary PMF, except that
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it applies to a new or revised
conditional model.
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As such, its entries
must also sum to 1.
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Now the random variable X has
a certain mean, expected
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value, which is defined
the usual way.
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In the conditional model, the
random variable X will also
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have a mean.
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It is called the conditional
mean or the conditional
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expectation.
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And it is defined the same way
as in the original case,
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except that now the calculation
involves the
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conditional probabilities,
or the conditional PMF.
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Finally, as we discussed some
time ago, a conditional
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probability model is just
another probability model,
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except that it applies
to a new situation.
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So any fact about probability
models--
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any theorem that we derive--
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must remain true in the
conditional model as well.
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As an example, the expected
value rule will have to remain
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true in the conditional model,
except that, of course, in the
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conditional model, we will have
to use the conditional
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probabilities instead of the
original probabilities.
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So to summarize, conditional
models and conditional PMFs
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are just like ordinary models
and ordinary PMFs, except that
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probabilities are replaced
throughout by conditional
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probabilities.
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Let us now look at an example.
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Consider a random variable,
which in this case, is
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uniform, takes values
from 1 up to 4.
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So each one of the possible
values has
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probability 1 over 4.
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For this random variable, we
can calculate the expected
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value, which by symmetry is
going to be the midpoint.
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So it is equal to 2 and 1/2.
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We can also calculate
the variance.
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And here we can apply
the formula that we
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have derived earlier--
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1/2 times b minus a times
b minus a plus 2.
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And in this case, it's 1 over 12
times b minus a is 4 minus
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1, which is 3.
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And the next term is 5.
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And after we simplify,
this is 5 over 4.
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Suppose that now somebody tells
us that event A has
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occurred, where event A is that
the random variable X
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takes values in the
range 2, 3, 4.
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What happens now?
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What is the conditional PMF?
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In the conditional model, we are
told that the value of 1
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did not occur, so this
probability is going to be 0.
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The other three values
are still possible.
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What are their conditional
probabilities?
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Well, these three values had
equal probabilities in the
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original model, so they should
have equal probabilities in
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the conditional model as well.
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And in order for probabilities
to sum to 1, of course, these
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probabilities will
have to be 1/3.
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So this is the conditional PMF
of our random variable, given
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this new or additional
information about the outcome.
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The expected value of the
random variable X in the
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conditional universe--
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that is, the conditional
expectation--
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is just the ordinary expectation
but applied to the
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conditional model.
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In this conditional model, by
symmetry, the expected value
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will have to be 3, the midpoint
of the distribution.
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And we can also calculate the
conditional variance.
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This is a notation that we have
not yet defined, but at
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this point, it should
be self-explanatory.
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It is just the variance of
X but calculated in the
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conditional model using
conditional probabilities.
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We can calculate this variance
using once more the formula
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for the variance of a uniform
distribution, but we can also
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do it directly.
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So the variance is the expected
value of the squared
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distance from the mean.
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So with probability 1/3, our
random variable will take a
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value of 4, which is one unit
apart from the mean, or more
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explicitly, we have this term.
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With probability 1/3,
our random variable
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takes a value of 3.
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And with probability 1/3, our
random variable takes the
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value of 2.
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This term is 0.
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This is 1 times 1/3.
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From here we get another
1 times 1/3.
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So adding up, we obtain that
the variance is 2/3.
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Notice that the variance in
the conditional model is
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smaller than the variance that
we had in the original model.
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And this makes sense.
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In the conditional model, there
is less uncertainty than
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there used to be in the
original model.
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And this translates into a
reduction in the variance.
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To conclude, there is nothing
really different when we deal
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with conditional PMFs,
conditional expectations, and
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conditional variances.
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They are just like the ordinary
PMFs, expectations,
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and variances, except that we
have to use the conditional
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probabilities throughout
instead of the original
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probabilities.