WEBVTT
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Hi.
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In this problem, we're going to
be dealing with a variation
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of the usual coin-flipping
problem.
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But in this case, the bias
itself of the coin
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is going to be random.
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So you could think of it as, you
don't even know what the
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probability of heads
for the coin is.
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So as usual, we're still taking
one coin and we're
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flipping it n times.
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But the difference here is that
the bias is because it
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was random variable Q. And
we're told that the
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expectation of this bias is some
mu and that the variance
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of the bias is some sigma
squared, which
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we're told is positive.
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And what we're going to be
asked is find a bunch of
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different expectations,
covariances, and variances.
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And we'll see that this problem
gives us some good
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exercise in a few concepts, a
lot of iterated expectations,
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which, again, tells you that
when you take the expectation
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of a conditional expectation,
it's just the expectation of
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the inner random variable.
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The covariance of two random
variables is just the
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expectation of the product
minus the product of the
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expectations.
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Law of total variance is the
expectation of a variance, of
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a conditional variance plus the
variance of a conditional
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expectation.
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And the last thing, of course,
we're dealing with a bunch of
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Bernoulli random variables,
coin flips.
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So as a reminder, for a
Bernoulli random variable, if
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you know what the bias is, it's
some known quantity p,
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then the expectation of the
Bernoulii is just p, and the
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variance of the Bernoulli
is p times 1 minus p.
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So let's get started.
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The problem tells us that we're
going to define some
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random variables.
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So xi is going to be a Bernoulli
random variable for
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the i coin flip.
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So xi is going to be 1 if the i
coin flip was heads and 0 if
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it was tails.
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And one very important thing
that the problem states is
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that conditional on Q, the
random bias, so if we know
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what the random bias is, then
all the coin flips are
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independent.
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And that's going to be important
for us when we
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calculate all these values.
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OK, so the first thing that we
need to calculate is the
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expectation of each of these
individual Bernoulli random
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variables, xi.
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So how do we go about
calculating what this is?
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Well, the problem
gives us a int.
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It tells us to try using the law
of iterated expectations.
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But in order to use it, you need
to figure out what you
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need the condition on.
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What this y?
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What takes place in y?
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And in this case, a good
candidate for what you
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condition on would be
the bias, the Q that
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we're unsure about.
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So let's try doing that
and see what we get.
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So we write out the law of
iterated expectations with Q.
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So now hopefully, we can
simplify it with this
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inter-conditional
expectation is.
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Well, what is it really?
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It's saying, given what Q is,
what is the expectation of
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this Bernoulli random
interval xi?
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Well, we know that if we knew
what the bias was, then the
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expectation is just
the bias itself.
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But in this case, the
bias is random.
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But remember a conditional
expectation is
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still a random variable.
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And so in this case, this
actually just simplifies into
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Q. So whatever the bias is, the
expectation is just equal
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to the bias.
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And so that's what
it tells us.
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And this part is easy because
we're given that the
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expectation of q is mu.
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And then the problem also
defines the random variable x.
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X is the total number of heads
within the n tosses.
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Or you can think of it as a sum
of all these individual xi
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Bernoulli random variables.
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And now, what can
we do with this?
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Well we can remember that
linearity of expectations
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allows us to split
up this sum.
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Expectation of a sum, we could
split up into a sum of
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expectations.
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So this is actually just
expectation of x1 plus dot dot
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dot plus all the way to
expectation of xn.
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All right.
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And now, remember that we're
flipping the same coin.
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We don't know what the bias is,
but for all the n flips,
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it's the same coin.
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And so each of these
expectations of xi should be
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the same, no matter
what xi is.
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And each one of them is mu.
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We already calculated
that earlier.
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And there's 10 of them, so the
answer would be n times mu.
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So let's move on to part B.
Part B now asks us to find
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what the covariance is
between xi and xj.
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And we have to be a little bit
careful here because there are
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two different scenarios, one
where i and j are different
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indices, different tosses,
and another where i
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and j are the same.
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So we have to consider both
of these cases separately.
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Let's first do the case where
x and i are different.
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So i does not equal j.
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In this case, we can just apply
the formula that we
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talked about in the beginning.
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So this covariance is just equal
to the expectation of xi
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times xj minus the expectation
of xi times expectation of xj.
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All right, so we actually know
what these two are, right?
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Expectation of xi is mu.
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Expectation of xj is also mu.
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So this part is just
mu squared.
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But we need to figure out
what this expectation
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of xi times xj is.
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Well, the expectation of xi
times xj, we can again use the
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law of iterated expectations.
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So let's try conditioning
on cue again.
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And remember we said
that this second
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part is just mu squared.
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All right, well, how can
we simplify this
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inner-conditional expectation?
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Well, we can use the fact that
the problem tells us that,
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conditioned on Q, the tosses
are independent.
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So that means that, conditioned
on Q, xi and xj
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are independent.
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And remember, when random
variables are independent, the
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expectation of product, you
could simplify that to be the
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product of the expectations.
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And because we're in the
condition world on Q, you have
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to remember that it's going
to be a product of two
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conditional expectations.
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So this will be expectation of
xi given Q times expectation
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of xj given Q minus
mu squared still.
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All right, now what is this?
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Well the expectation of xi given
Q, we already argued
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earlier here that it should just
be Q. And then the same
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thing for xj.
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That should also be Q. So this
is just expectation of Q
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squared minus mu squared.
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All right, now if we look at
this, what is the expectation
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of Q squared minus mu squared?
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Well, remember mu is just,
we're told that mu is the
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expectation of Q. So what we
have is the expectation of Q
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squared minus the quantity
expectation of Q squared.
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And what is that, exactly?
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That is just the formula or
the definition of what the
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variance of Q should be.
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So this is, in fact, exactly
equal to the variance of Q,
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which we're told is
sigma squared.
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All right, so what we found is
that for i not equal to j, the
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coherence of xi and
xj is exactly
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equal to sigma squared.
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And remember, we're told that
sigma squared is positive.
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So what does that tell us?
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That tells us that xi and xj, or
i not equal to j, these two
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random variables
are correlated.
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And so, because they're
correlated, they can't be
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independent.
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Remember, if two intervals are
independent, that means
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they're uncorrelated.
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But the converse isn't true.
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But if we do know that two
random variables are
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correlated, that means that
they can't be independent.
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And now let's finish this by
considering the second case.
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The second case is when i
actually does equal j.
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And in that case, well, the
covariance of xi and xi is
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just another way of writing
the variance of xi.
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So covariance, xi, xi, it's
just the variance of xi.
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And what is that?
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That is just the expectation
of xi squared minus
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expectation of xi quantity
squared.
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And again, we know what
the second term is.
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The second term is expectation
of xi quantity squared.
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Expectation of xi we know from
part A is just mu, right?
00:10:26.540 --> 00:10:28.670
So that's just second term
is just mu squared.
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But what is the expectation
of xi squared?
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Well, we can think about
this a little bit more.
00:10:35.220 --> 00:10:40.020
And you can realize that xi
squared is actually exactly
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the same thing as just xi.
00:10:41.920 --> 00:10:45.150
And this is just a special case
because xi is a Bernoulli
00:10:45.150 --> 00:10:46.230
random variable.
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Because Bernoulli is
either 0 or 1.
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And if it's 0 and you square
it, it's still 0.
00:10:52.010 --> 00:10:54.380
And if it's 1 and you square
it, it's still 1.
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So squaring it doesn't
really doesn't
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actually change anything.
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It's exactly the same thing as
the original random variable.
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And so, because this is a
Bernoulli random variable,
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this is exactly just the
expectation of xi.
00:11:11.340 --> 00:11:13.880
And we said this part
is just mu squared.
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So this is just expectation of
xi, which we said was mu.
00:11:17.950 --> 00:11:21.730
So the answer is just
mu minus mu squared.
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OK, so this completes part B.
And the answer that we wanted
00:11:31.880 --> 00:11:38.190
was that in fact, xi and xj are
in fact not independent.
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Right.
00:11:39.130 --> 00:11:45.960
So let's write down some facts
that we'll want to remember.
00:11:45.960 --> 00:11:51.610
One of them is that expectation
of xi is mu.
00:11:51.610 --> 00:11:56.660
And we also want to remember
what this covariance is.
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The covariance of xi and xj is
equal to sigma squared when i
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does not equal j.
00:12:06.470 --> 00:12:10.570
So we'll be using these
facts again later.
00:12:10.570 --> 00:12:18.780
And the variance of xi is equal
to mu minus mu squared.
00:12:22.120 --> 00:12:27.830
So now let's move on to the last
part, part C, which asks
00:12:27.830 --> 00:12:34.550
us to calculate the variance
of x in two different ways.
00:12:34.550 --> 00:12:39.110
So the first way we'll
do it is using the
00:12:39.110 --> 00:12:41.830
law of total variance.
00:12:41.830 --> 00:12:47.470
So the law of total variance
will tell us that we can write
00:12:47.470 --> 00:12:51.940
the variance of x as a sum
of two different parts.
00:12:51.940 --> 00:12:56.240
So the first is variance of x
expectation of the variance of
00:12:56.240 --> 00:13:03.740
x conditioned on something
plus the variance of the
00:13:03.740 --> 00:13:07.320
initial expectation of x
conditioned on something.
00:13:07.320 --> 00:13:10.030
And as you might have guessed,
what we're going to condition
00:13:10.030 --> 00:13:16.330
on is Q.
00:13:16.330 --> 00:13:18.670
Let's calculate what these
two things are.
00:13:18.670 --> 00:13:21.170
So let's do the two
terms separately.
00:13:21.170 --> 00:13:23.470
What is the expectation
of the conditional
00:13:23.470 --> 00:13:26.490
variance of x given Q?
00:13:29.750 --> 00:13:33.550
Well, what is--
00:13:33.550 --> 00:13:36.140
this, we can write out x.
00:13:36.140 --> 00:13:41.880
Because x, remember, is just
the sum of a bunch of these
00:13:41.880 --> 00:13:43.270
Bernoulli random variables.
00:13:46.290 --> 00:13:50.380
And now what we'll do was, well,
again, use the important
00:13:50.380 --> 00:13:54.900
fact that the x's, we're told,
are conditionally independent,
00:13:54.900 --> 00:13:56.710
conditional on Q.
00:13:56.710 --> 00:14:00.450
And because they're independent,
remember the
00:14:00.450 --> 00:14:03.560
variance of a sum is not the
sum of the variance.
00:14:03.560 --> 00:14:06.730
It's only the sum of the
variance if the terms in the
00:14:06.730 --> 00:14:08.480
sum are independent.
00:14:08.480 --> 00:14:10.880
In this case, they are
conditionally independent
00:14:10.880 --> 00:14:15.730
given Q. So we can in fact split
this up and write it as
00:14:15.730 --> 00:14:20.340
the variance of x1 given Q
plus all the way to the
00:14:20.340 --> 00:14:30.980
variance of xn given Q.
00:14:30.980 --> 00:14:33.960
And in fact, all these
are the same, right?
00:14:33.960 --> 00:14:39.530
So we just have n copies of the
variance of, say, x1 given
00:14:39.530 --> 00:14:43.310
Q. Now, what is the variance
of x1 given Q?
00:14:43.310 --> 00:14:46.770
Well, x1 is just a Bernoulli
random variable.
00:14:46.770 --> 00:14:51.620
But the difference is that for
x, we don't know what the bias
00:14:51.620 --> 00:14:54.060
or what the Q is.
00:14:54.060 --> 00:14:57.910
Because it's some
random bias Q
00:14:57.910 --> 00:15:01.010
But just like we said earlier
in part A, when we talked
00:15:01.010 --> 00:15:07.640
about the expectation of x1
given Q, this is actually just
00:15:07.640 --> 00:15:13.250
Q times 1 minus Q. Because if
you knew what the bias were,
00:15:13.250 --> 00:15:14.810
it would be p times 1 minus p.
00:15:14.810 --> 00:15:16.860
So the bias times 1
minus the bias.
00:15:16.860 --> 00:15:19.190
But you don't know what it is.
00:15:19.190 --> 00:15:21.060
But if you did, it
would just be q.
00:15:21.060 --> 00:15:23.870
So what we do is we just plug
in Q, and you get Q
00:15:23.870 --> 00:15:26.770
times 1 minus 2.
00:15:26.770 --> 00:15:36.110
All right, and now this
is expectation of n.
00:15:36.110 --> 00:15:38.960
I can pull out the n.
00:15:38.960 --> 00:15:43.470
So it's n times the expectation
of Q minus Q
00:15:43.470 --> 00:15:51.090
squared, which is just n times
expectation Q, we can use
00:15:51.090 --> 00:15:55.450
linearity of expectations again,
expectation of Q is mu.
00:15:55.450 --> 00:16:00.540
And the expectation of Q 2
squared is, well, we can do
00:16:00.540 --> 00:16:01.230
that on the side.
00:16:01.230 --> 00:16:08.840
Expectation of Q squared is
the variance of Q plus
00:16:08.840 --> 00:16:14.230
expectation of Q quantity
squared.
00:16:14.230 --> 00:16:22.120
So that's just sigma squared
plus mu squared.
00:16:22.120 --> 00:16:27.810
And so this is just going to
be then minus sigma squared
00:16:27.810 --> 00:16:29.060
minus mu squared.
00:16:32.080 --> 00:16:33.820
All right, so that's
the first term.
00:16:33.820 --> 00:16:35.950
Now let's do the second term.
00:16:35.950 --> 00:16:43.720
The variance the conditional
expectation of x given Q. And
00:16:43.720 --> 00:16:52.740
again, what we can do is we can
write x as the sum of all
00:16:52.740 --> 00:16:55.435
these xi's.
00:16:59.270 --> 00:17:04.730
And now we can apply linearity
of expectations.
00:17:04.730 --> 00:17:08.705
So we would get n times one
of these expectations.
00:17:13.440 --> 00:17:18.530
And remember, we said earlier
the expectation of x1 given Q
00:17:18.530 --> 00:17:23.720
is just Q. So it's the variance
of n times Q.
00:17:23.720 --> 00:17:26.375
And remember now, n is just--
00:17:26.375 --> 00:17:27.460
it's not random.
00:17:27.460 --> 00:17:29.680
It's just some number.
00:17:29.680 --> 00:17:32.070
So when you pull it out of a
variance, you square it.
00:17:32.070 --> 00:17:36.290
So this is n squared times
the variance of Q.
00:17:36.290 --> 00:17:39.130
And the variance of Q we're
given is sigma squared.
00:17:39.130 --> 00:17:42.660
So this is n squared times
sigma squared.
00:17:45.280 --> 00:17:47.860
So the final answer is
just a combination
00:17:47.860 --> 00:17:49.250
of these two terms.
00:17:49.250 --> 00:17:54.290
This one and this one.
00:17:54.290 --> 00:17:56.010
So let's write it out.
00:17:56.010 --> 00:17:59.295
The variance of x, then,
is equal to--
00:18:02.790 --> 00:18:04.580
we can combine terms
a little bit.
00:18:04.580 --> 00:18:08.010
So the first one, let's
take the mus and
00:18:08.010 --> 00:18:08.730
we'll put them together.
00:18:08.730 --> 00:18:11.325
So it's n mu minus mu squared.
00:18:15.830 --> 00:18:22.660
And then we have n squared times
sigma squared from this
00:18:22.660 --> 00:18:28.520
term and minus n times sigma
squared from this term.
00:18:28.520 --> 00:18:34.450
So it would be n squared minus
n times sigma squared, or n
00:18:34.450 --> 00:18:38.400
times n minus 1 times
sigma squared.
00:18:38.400 --> 00:18:40.970
So that is the final answer
that we get for
00:18:40.970 --> 00:18:42.220
the variance of x.
00:18:45.030 --> 00:18:47.450
And now, let's try doing
it another way.
00:18:51.800 --> 00:18:53.960
So that's one way of doing it.
00:18:53.960 --> 00:18:57.140
That's using the law of total
expectations and conditioning
00:18:57.140 --> 00:19:05.880
on Q. Another way of finding
the variance of x is to use
00:19:05.880 --> 00:19:11.330
the formula involving
covariances, right?
00:19:11.330 --> 00:19:18.652
And we can use that because x is
actually a sum of multiple
00:19:18.652 --> 00:19:23.590
random variables
x1 through xn.
00:19:23.590 --> 00:19:40.780
And the formula for this is, you
have n variance terms plus
00:19:40.780 --> 00:19:44.110
all these other ones.
00:19:44.110 --> 00:19:48.140
Where i is not equal to j, you
have the covariance terms.
00:19:48.140 --> 00:19:51.770
And really, it's just, you can
think of it as a double sum of
00:19:51.770 --> 00:19:59.150
all pairs of xi and xj where if
i and j happen just to be
00:19:59.150 --> 00:20:02.710
the same, that it simplifies
to be just the variance.
00:20:02.710 --> 00:20:06.240
Now, so we pulled theses n terms
out because they are
00:20:06.240 --> 00:20:10.770
different than these because
they have a different value.
00:20:10.770 --> 00:20:14.060
And now fortunately, we've
already calculated what these
00:20:14.060 --> 00:20:16.690
values are in part B. So we
can just plug them them.
00:20:16.690 --> 00:20:18.890
All the variances
are the same.
00:20:18.890 --> 00:20:21.300
And there's n of them,
so we get n times the
00:20:21.300 --> 00:20:22.260
variance of each one.
00:20:22.260 --> 00:20:26.960
The variance of each one we
calculated already was mu
00:20:26.960 --> 00:20:29.790
minus mu squared.
00:20:29.790 --> 00:20:32.630
And then, we have all
the terms were i is
00:20:32.630 --> 00:20:34.210
not equal to j.
00:20:34.210 --> 00:20:39.650
Well, there are actually n
squared minus n of them.
00:20:39.650 --> 00:20:44.040
So because you can take any one
of the n's to be the first
00:20:44.040 --> 00:20:48.110
to be i, any one of
the n to be j.
00:20:48.110 --> 00:20:49.890
So that gives you
n squared pairs.
00:20:49.890 --> 00:20:52.590
But then you have to subtract
out all the ones where i and j
00:20:52.590 --> 00:20:53.190
are the same.
00:20:53.190 --> 00:20:54.320
And there are n of them.
00:20:54.320 --> 00:20:59.250
So that leaves you with n
squared minus n of these pairs
00:20:59.250 --> 00:21:01.600
where i is not equal to j.
00:21:01.600 --> 00:21:04.130
And the coherence for this case
where i is not equal to
00:21:04.130 --> 00:21:08.176
j, we also calculated in part B.
That's just sigma squared.
00:21:08.176 --> 00:21:13.050
All right, and now if we compare
these two, we'll see
00:21:13.050 --> 00:21:15.610
that they are proportionally
exactly the same.
00:21:18.510 --> 00:21:23.700
So we've use two different
methods to calculate the
00:21:23.700 --> 00:21:27.510
variance, one using this
summation and one using the
00:21:27.510 --> 00:21:29.860
law of total variance.
00:21:29.860 --> 00:21:33.040
So what do we learn
from this problem?
00:21:33.040 --> 00:21:37.430
Well, we saw that first of all,
in order to find some
00:21:37.430 --> 00:21:40.940
expectations, it's very useful
to use law of iterated
00:21:40.940 --> 00:21:41.700
expectations.
00:21:41.700 --> 00:21:44.620
But the trick is to figure out
what you should condition on.
00:21:44.620 --> 00:21:47.780
And that's kind of an
art that you learn
00:21:47.780 --> 00:21:49.230
through more practice.
00:21:49.230 --> 00:21:52.920
But one good rule of thumb is,
when you have kind of a
00:21:52.920 --> 00:21:57.650
hierarchy or layers of
randomness where one layer of
00:21:57.650 --> 00:22:00.640
randomness depends
on the randomness
00:22:00.640 --> 00:22:01.960
of the layer above--
00:22:01.960 --> 00:22:05.780
so in this case, whether or
not you get heads or tails
00:22:05.780 --> 00:22:09.600
depends on, that's random, but
that depends on the randomness
00:22:09.600 --> 00:22:12.040
on the level above, which
was the random
00:22:12.040 --> 00:22:14.150
bias of the coin itself.
00:22:14.150 --> 00:22:19.410
So the rule of thumb is, when
you want to calculate the
00:22:19.410 --> 00:22:23.360
expectations for the layer where
you're talking about
00:22:23.360 --> 00:22:27.710
heads or tails, it's useful to
condition on the layer above
00:22:27.710 --> 00:22:30.590
where that is, in this case,
the random bias.
00:22:30.590 --> 00:22:34.430
Because once you condition on
the layer above, that makes
00:22:34.430 --> 00:22:36.210
the next level much simpler.
00:22:36.210 --> 00:22:39.830
Because you kind of assume that
you know what all the
00:22:39.830 --> 00:22:42.650
previous levels of randomness
are, and that helps you
00:22:42.650 --> 00:22:47.480
calculate what the expectation
for this current level.
00:22:47.480 --> 00:22:52.180
And the rest of the problem was
just kind of going through
00:22:52.180 --> 00:22:54.160
exercises of actually
applying the--