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ALBERT MEYER: In
the last lecture,

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we spent time talking about
the mean, or expectation,

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and its properties, most
important one being linearity.

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But let's step back
now and think about,

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what is it that the mean means?

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Why we care about it?

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We have this intuitive idea that
if you do things long enough,

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if you keep experimenting
with the same random variable

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collecting its values,
its long run average will

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be about the same as its mean.

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Now, we're going to try
to make that more precise.

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So we're going to talk
about the topic of deviation

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from the mean, or
as I like to say,

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what does the mean really mean?

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Why do we care about it?

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Well, let's look at
an example that's

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familiar to get a grip
on the specific ideas

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that we're interested in.

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So suppose I toss a
fair coin 101 times.

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Then, I know that the expected
number, since all the values

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from zero through 101 are
possible, and the middle value

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is the expectation,
it's 50 and 1/2 heads.

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Now, I'm never going to get
exactly 50 and 1/2 heads.

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The probability in 101 flips
of getting 50 and 1/2 heads

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is zero because there's
no way to flip 1/2 a head.

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So you don't expect the
expectation in that sense.

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No given measurement,
no given experiment

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is going to yield
the expectation.

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The expectation is this
thing that we expect

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to come out on the average.

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Well, we can ask,
what's the probability

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of getting as close as you could
hope to get to the expectation?

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Namely, what's the probability
of getting exactly 50 heads?

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And it's about 1/13.

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Or if you ask, what's
the probability

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of getting either
50 or 51 heads,

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being within plus or minus
one of the expectation?

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It's about 1/7.

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OK, let's flip more coins
and see what happens.

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This time I'm going
to flip 1001 coins.

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And again, the expected number
of heads is 500 and 1/2,

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which I'll never get exactly.

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The probability of getting
exactly 500 heads is 1/39,

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and the probability of getting
within one of the expectation,

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that is either 500 or
501 heads, is about 1/19.

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Now, these numbers have gone
down from the previous numbers.

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Remember, it was about 1/7
and we've gone down to 1/19.

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So it's actually we're
less likely to be

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within a fixed distance, within
one of the expectation when

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we flip more coins.

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So as the number of tosses
grows, the number of heads

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gets less likely to be
within any given fixed

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distance of the mean.

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But things get better when we
start looking at percentages.

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So what's the
probability of being

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within 1% of the mean
if I toss 1,001 coins?

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Well, 1% of 1,001 is about
10, so we're talking about 1%

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of the 1,001.

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And the probability of
being within 10 of 500.5,

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that is to say the probability
of being within 510 and 490,

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is about 0.49.

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It's almost 50-50, which
is not really so bad.

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So we have a 50/50
chance of actually

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being within 1% of the expected
number when I flip 1,001 coins.

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So what we can start
to say is that when

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we're trying to give
the meaning to the mean,

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if I let u be the standard
abbreviation for expectation

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of R-- I'm doing that just
so it'll fit on the slide

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nicely in formulas, so mu
is the expectation of R--

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the basic question we're
asking is two basic questions.

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One is, what's the probability
that the random variable is far

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from its mean, mu?

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You could phrase that as,
what's the probability

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that the distance from R to
mu, the absolute value of R

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minus mu is greater
than some amount, x.

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And the second question
that we want to ask

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is, what's the
average deviation?

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What's the expectation of the
distance between R minus mu?

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What's the expected
value of r minus mu?

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Now, of course,
we're trying to make

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sense of the meaning
of the expectation,

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in terms of the expectation
of the distance between R

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and this expectation.

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So there's a little bit
of circularity there,

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but let's live with
it and proceed.

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Let's look at example to
crystallize the ideas a little.

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Let's look at two dice
with the same mean.

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The green die is going to be
a standard fair die, in which

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each of the numbers
one through six

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has an equal probability
of showing up,

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and its expected
value is exactly

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going to be the midpoint between
one and six, or 3 and 1/2.

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Now, suppose I look
at a loaded die,

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die two, which only
throws a one or a six,

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but with equal probability.

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Then, it's expectation is also 3
and 1/2, by the same reasoning.

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So here are these
two different die.

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One takes the values one
through six equally likely,

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and the other takes only
the two values one and six,

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but they have the
same expectation.

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So how do I capture
the difference?

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Well, if I look at the expected
distance of the fair die

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to its mean, I claim
it's one and a half.

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But the expected distance
of the loaded die

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from its mean-- same mean
remember, 3 and 1/2--

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is actually 2 and 1/2.

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In fact, the second die is
always exactly 2 and 1/2

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from its expected value.

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Let's look at the PDFs to
get a grip on understanding

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what's going on.

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So here's the PDF
for the fair die.

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Over one through six
the probability is 1/6,

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so each of those green
spikes, columns, is 1/6 high.

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And their total
is the probability

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that the fair die takes one of
those values one through six

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with equal likelihood.

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Now, the expected
value is exactly

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in the middle at 3 and 1/2.

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And the average distance
of these points--

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well, you can see that a
third of the time, you're

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at distance 1/2, a
third of the time, that

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is when you take
the values 2 and 5,

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you are a distance
exactly 1 and 1/2.

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And another third of the time,
you're at distance 2 and 1/2

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when you take one and six.

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And that averages out to the
middle value of 1 and 1/2.

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So the expected deviation,
the expected distance,

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of the fair die from
its mean is 1 and 1/2.

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On the other hand, for the
loaded die, as we said,

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it's always exactly 2
and 1/2 from its expected

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value, which means its expected
value is also 2 and 1/2.

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So we can start to see the
difference between these two

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distributions and
these two kinds of die.

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Even though they have the
same expectation, one of them

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is more likely and has a
greater expected distance

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from its mean.

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And the moral of
this short piece

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is that the mean alone
is not a good predictor

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of a random variable's
behavior, as you might suppose.

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One parameter, one
number is not going

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to capture the shape of
a PDF, which gives you

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more full information about
the distribution of values

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of a random variable.

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And we need some
more information

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than just the expectation.

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An especially, valuable
extra piece of information

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that's still well less
than the overall shape

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of the PDF of the
random variable,

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is knowing its probable
deviation from its mean.