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Hi.
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In this problem, we'll get a
chance to see the usefulness
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of conditioning in helping us
to calculate quantities that
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would otherwise be difficult
to calculate.
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Specifically, we'll be using
the law of iterated
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expectations and the law
of total variance.
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Before we get started, let's
just take a quick moment to
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interpret what these two
laws are saying.
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Really, what it's saying is,
in order to calculate the
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expectation or the variance of
some random variable x, if
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that's difficult to do,
we'll instead attack
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this problem in stages.
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So the first stage is, we'll
condition on some related
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random variable, y.
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And the hope is that by
conditioning on this and
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reducing it to this conditional
universe, the
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expectation of x will be
easier to calculate.
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Now, recall that this
conditional expectation is
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really a random variable, which
is a function of the
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random variable y.
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So what we've done is we first
average out x given some y.
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What remains is some new random
variable, which is a
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function of y.
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And now, what we have is
randomness in y, which will
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then average out again to get
the final expectation of x.
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OK, so in this problem, we'll
actually see an example of how
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this plays out.
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One more thing before we get
started that's useful to
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recall is if y is a uniform
random variable, distributed
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between a and b, then the
variance of y is b minus a
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squared over 12, and the
expectation of y is just a
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midpoint, a plus b over 2.
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All right, so let's get started
on the problem.
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So what we have is we have a
stick of some fixed length, l,
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and what we do is we break
it uniformly at random.
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So what we do is we choose a
point uniformly at random
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along this stick.
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And we break it there, and then
we keep the left portion
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of that stick.
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So let's call the length of this
left portion after the
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first break random variable y.
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So it's random because
the point where
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we break it is random.
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And then what we do is we
repeat this process.
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We'll take this left side of
the stick that's left.
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And we'll pick another point,
uniformly at random, along
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this left remaining side.
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And we'll break it again,
and keep the left
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side of that break.
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And we'll call that the length
of the final remaining piece,
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x, which again is random.
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The problem is really asking
us to calculate the
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expectation of variance of x.
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So at first, it seems difficult
to do, because the
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expectation and variance of x
depends on where you break it
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the second time and also where
you break it the first time.
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So let's see if conditioning
can help us here.
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So the first thing that we'll
notice is that, if we just
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consider y, the length of the
stick after the first break,
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it's actually pretty easy to
calculate the expectation and
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variance of y.
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Because y, when you think about
it, is actually just the
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simple uniform in a variable,
uniformly distributed between
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0 and l, the length
of the stick.
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And this is because we're told
that we choose the point of
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the break uniformly at random
between 0 and l.
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And so wherever we choose it,
that's going to be the length
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of the left side of the stick.
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And so because of this, we know
that the expectation of y
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is just l/2, and the variance
of y is l squared over 12.
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But unfortunately, calculating
the expectation variance of x
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is not quite as simple, because
x isn't just uniformly
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distributed between 0 and
some fixed number.
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Because it's actually uniformly
distributed between
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0 and y, wherever the
first break was.
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But where the first break
is is random too.
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And so we can't just say that
x is a uniformly distributed
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random variable.
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So what do we do instead?
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Well, we'll make the nice
observation that let's pretend
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that we actually
know what y is.
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If we knew what y was, then
calculating the expectation of
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x would be simple, right?
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So if we were given that y is
just some little y, then x
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would in fact just be uniformly
distributed between
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0 and little y.
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And then if that's the case,
then our calculation is
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simple, because the expectation
of x would just be
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y/2, and the variance would
just be y squared over 12.
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All right, so let's make that
a little bit more formal.
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What we're saying is that the
expectation of x, If we knew
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what y was, would just be y/2.
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And the variance of x If we knew
what y was would just be
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y squared over 12.
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All right, so notice
what we've done.
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We've taken the second stage and
we've said, let's pretend
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we know what happens in the
first stage where we break it.
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And we know what y, the
first break, was.
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Then the second stage becomes
simple, because the average of
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x is just going to
be the midpoint.
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Now what we do to calculate the
actual expectation of x,
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well, we'll invoke the law
of iterated expectations.
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So expectation of x is
expectation of the conditional
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expectation of x given 1, which
in this case is just
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expectation of y/2.
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And we know what the expectation
of y is.
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It's l/2.
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And so this is just l/4.
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l/4.
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All right, and so notice
what we've done.
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We've taken this calculation
and done it in stages.
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So we assume we know where
the first break is.
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Given that, the average location
of the second break
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becomes simple.
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It's just in the midpoint.
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And then, we move up to
the higher stage.
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And that now we average out
over where the first break
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could have been.
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And that gives us our
final answer.
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And notice that this actually
makes sense, if we just think
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about it intuitively, because
on average, the first break
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will be somewhere
in the middle.
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And then that will leave us with
half the stick left, and
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we break it again.
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On average, that will leave
us with another half.
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So on average, you get a quarter
of the original stick
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left, which makes sense.
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All right, so that's the first
part, where we use the law of
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iterated expectations.
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Now, let's go to part B, where
we're actually asked to find
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the variance.
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The variance is given by the
law of total variance.
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So let's do it in stages.
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We'll first calculate the first
term, the expectation of
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the conditional variance.
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Well, what is the expectation
of the conditional variance?
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We've already calculated
out what this
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conditional variance is.
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The conditional variance
is y squared over 12.
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So let's just plug that in.
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It's expectation of
y squared over 12.
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All right, now this looks like
it could be a little difficult
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to calculate.
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But let's just first
pull out the 1/12.
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And then remember, one way to
calculate the expectation of
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the square of a random variable
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is to use the variance.
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So recall that the variance of
any random variable is just
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expectation of the square
minus the square of the
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expectation.
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So if we want to calculate the
expectation of the square, we
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can just take the variance
and add the square of the
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expectation.
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So this actually we can
get pretty easily.
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It's actually just the variance
of y plus the square
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of the expectation of y.
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And we know what these
two terms are.
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The variance of y is
l squared over 12.
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And the expectation
of y is l/2.
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So when you square that, you
get l squared over 4.
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So l squared over 12 plus l
squared over 4 gives you l
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squared over 3.
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And you get that the first term
is l squared over 36.
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All right, now let's calculate
the second term.
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Second term is the variance of
the conditional expectation.
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So the variance of expectation
of x given y.
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Well, what is the expectation
of x given y?
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We've already calculated that.
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That's y/2.
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So what we really want is
the variance of y/2.
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And remember, when you have a
constant inside the variance,
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you pull it out but
you square it.
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So what you get is 1/4 the
variance of y, which we know
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that the variance of y
is l squared over 12.
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So we get that this is
l squared over 48.
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OK, so we've calculated
both terms of
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this conditional variance.
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So all we need to do to
find the final answer
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is just to add them.
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So it's l squared over 36
plus l squared over 48.
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And so, the final answer is
7 l squared over 144.
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OK, and so this is the first,
the expectation of x, maybe
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you could have guessed
intuitively.
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But the variance of x is not
something that looks like
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something that you could
have calculated off
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the top of your head.
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And so I guess the lesson from
this example is that it is
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often very helpful if you
condition on some things,
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because it allows you to
calculate things in stages and
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build up from the bottom.
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But it's important to note that
the choice of what you
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condition on-- so the choice
of y-- is actually very
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important, because you could
choose lots of other y's that
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wouldn't actually
help you at all.
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And so how to actually choose
this y is something that you
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can learn based on just
having practiced with
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these kinds of problems.
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So again, the overall lesson
is, conditioning can often
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help when you calculate
these problems.
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And so you should look
to see if that could
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be a possible solution.
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So I hope that was helpful,
and see you next time.
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