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In this problem, we're given a
collection of 10 variables, x1
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through x10, where each i, xi,
is a uniform random variable
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between 0 and 1.
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So each i is uniform between 0
and 1, and all 10 variables
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are independent.
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And we'd like to develop a bound
on the probability that
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some of the 10 variables, 1 to
10, being greater than 7 using
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different methods.
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So in part A we'll be using
the Markov's inequality
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written here.
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That is, if we have a random
variable, positive random
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variable x, the probability x is
greater than a, where a is
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again some positive number, is
bounded above by the expected
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value of x divided by a.
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And let's see how that works
out in our situation.
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In our situation, we will call
x the summation of i equal to
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1 to 10xi, and therefore, E of
x is simply 10 times E of x1,
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the individual ones, and
this gives us 5.
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Here we use used the linearity
of expectation such that the
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expectation of the sum of the
random variable is simply the
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sum of the expectations.
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Now, we can invoke Markov's
Inequality.
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It says x greater
or equal to 7.
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This is less than E of x over 7,
and this gives us 5 over 7.
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For part B, let's see if we can
improve the bound we got
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in part A using the Chebyshev
inequality, which takes into
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account the variance of
random variable x.
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Again, to refresh you on this,
the Chebyshev Inequality says
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the probability that x deviates
from its mean E of x,
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by more than a is bound above
by the variance of x divided
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by a squared.
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So we have to actually do some
work to transform the
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probability we're interested
in, which is x greater or
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equal to 7, into the form that's
convenient to use using
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the Chebyshev Inequality.
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To do so, we'll rewrite this
probability as the probability
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of x minus 5 greater or equal
to 2 simply by moving 5 from
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the right to the left.
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The reason we chose 5 is because
5 is equal to the
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expected value of x from part
A as we know before.
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And in fact, this quantity is
also equal to the probability
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that x minus 5 less or
equal to negative 2.
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To see why this is true, recall
that x is simply the
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summation of the xi's, the 10
xi's, and each xi is a uniform
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random variable between
0 and 1.
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And therefore, each xi, the
distribution of which is
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symmetric around its mean 1/2.
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So we can see that after we
add up all the xi's, the
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resulting distribution
x is also symmetric
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around its mean 5.
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And as a result, the probability
of x minus 5
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greater than 2 is now equal to
the probability that x minus 5
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less than negative 2.
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And knowing these two, we can
then say they're both equal to
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1/2 the probability x minus 5
absolute value greater or
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equal to 2, because this term
right here is simply the sum
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of both terms here and here.
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At this point, we have
transformed the probability of
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x greater or equal to 7 into the
form right here, such that
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we can apply the Chebyshev's
Inequality basically directly.
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And we'll write the probably
here being less than or equal
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to 1/2 times, applying the
Chebyshev Inequality, variance
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of x divided by 2 squared.
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Now, 2 is the same as a
right here, and this
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gives us 1/8 times--
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now, the variance of x, we know
is 10 times the variance
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of a uniform random variable
between 0 and 1, which is
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1/12, and that gives us 5/48.
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Now, let's compare this with
the number we got earlier
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using the Markov Inequality,
which was 5/7.
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We see that 5/48 is much
smaller, and this tells us
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that, at least for this example,
using the Chebyshev
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Inequality combined with the
information of the variance of
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x, we're able to get a stronger
upper bound on the
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probability of the event that
we're interested in.
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Now, in part B, we saw that
by using the additional
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information of the variance
combined with the Chebyshev
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Inequality, we can improve upon
bound given by Markov's
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Inequality.
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Now, in part C, we'll use
a somewhat more powerful
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approach in addition to the
Chebyshev Inequality, the
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so-called central
limit theorem.
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Let's see if we can even
get a better bound.
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To remind you what a central
limit theorem is, let's say we
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have a summation of i equal
to 1 to some number n of
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independent and identically
distributed
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random variables xi.
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Now, the central limit theorem
says the following.
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We take the sum right here, and
subtract out its means,
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which is E of the same
summation, and further, we'll
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divide out, what we call
normalize, by the standard
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deviation of the summation.
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In other words, the square
root of the variance
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of the sum of xi.
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So if we perform this procedure
right here, then as
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the number of terms in the sums
going to infinity, here
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as in goes to infinity, we will
actually see that this
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random variable will converge
in distribution in some way
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that will eventually look like
a standard normal random
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variable with means 0 and 1.
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And since we know how the
distribution of a standard
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normal looks like, we can go to
table and look up certain
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properties of the resulting
distribution.
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So that is a plan to do.
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So right now, we have
about 10 variables.
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It's not that many compared to
a huge numbering, but again,
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if we believe it's a good
approximation, we can get some
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information out of it by using
the central limit theorem.
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So we are interesting knowing
that probability summation of
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i equal to 1 to 10 x1 greater
or equal to 7.
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We'll rewrite this as 1 minus
the probability the summation
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i equal to 1 to 10, and
xi less equal to 7.
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Now, we're going to apply
the scaling to the
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summation right here.
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So this is equal to 1 minus the
probability summation i
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equal to 1 to 10xi minus 5.
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Because we know from previous
parts that 5 is the expected
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value of the sum right
here, and divided by
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square root of 10/12.
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Again, earlier we know that
10/12 is the variance of the
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sum of xi's.
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And we'll do the same on the
other side, writing it 7 minus
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5 divided by square
root of 10/12.
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Now, if we compute out the
quantity right here, we know
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that this quantity is roughly
2.19, and by the central limit
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theorem, if we believe 10 is a
large enough number, then this
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will be roughly equal to 1 minus
the CDF of a standard
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normal evaluated at 2.19.
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And we could look up the number
in the table, and this
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gives us number roughly,
0.014.
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Now let's do a quick summary of
what this problem is about.
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We're asked to compute the
probability of x greater or
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equal to 7, where x is the sum
of 10 uniform random variables
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between 0 and 1, so
we'll call it xi.
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We know that because each
random variable has
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expectation 1/2, adding 10
of them up, gives us
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expectation of 5.
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So this is essentially asking,
what is the chance that x is
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more than two away from
its expectation?
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So if this is a real line, and
5 is here, maybe x has some
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distribution around 5, so the
center what the expected value
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is at 5, we wonder how likely is
it for us to see something
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greater than 7?
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Now, let's see where do we land
on the probably spectrum
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from 0 to 1.
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Well, without using any
information, we know the
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probability cannot be greater
than 1, so a trivial upper
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bound for the probability
right here will be 1.
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Well, for the first part we use
Markov's Inequality and
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that gives us some number, which
is roughly equal to 0.7.
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In fact, we got number 5/7,
and this is from Markov's
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Inequality.
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Oh, it's better than 1, already
telling us it cannot
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be between 0.7 and 1, but
can we do better?
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Well, the part B, we see that
all the way, using the
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additional information variance,
we can get this
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number down to 5/48, which
is roughly 0.1.
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Already, that's much
better than 0.7.
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Can we even do better?
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And this is the Chebyshev,
and it turns out we
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can indeed do better.
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Using the central limit theorem,
we can squeeze this
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number all the way down to
0.014, almost a 10 times
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improvement over the
previous number.
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This is from central
limit theorem.
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As we can see, by using
different bounding techniques,
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we can progressively improve the
bound on the probability
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of x exceeding 7, and from this
problem we learned that
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even with 10 variables, the
truth is more like this, which
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says that the distribution of
x concentrates very heavily
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around 5, and hence, the
probability of x being greater
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or equal to 7 could be much
smaller than one might expect.
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