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visit MIT OpenCourseWare at ocw.mit.edu. PHILIPPE RIGOLLET: OK, so the
course you're currently sitting in is 18.650. And it's called
Fundamentals of Statistics. And until last spring, it
was still called Statistics for Applications. It turned out that really, based
on the content, "Fundamentals of Statistics" was a
more appropriate title. I'll tell you a
little bit about what we're going to be covering
in class, what this class is about, what it's not about. I realize there's
several offerings in statistics on campus. So I want to make sure that
you've chosen the right one. And I also understand
that for some of you, it's a matter of scheduling. I need to actually
throw out a disclaimer. I tend to speak too fast. I'm aware that. Someone in the back, just
do like that when you have no idea what I'm saying. Hopefully, I will repeat
myself many times. So if you average
over time, you'll see that statistics
will tell you that you will get the right
message that I was actually trying to stick to send. All right, so what are
the goals of this class? The first one is basically
to give you an introduction. No one here is expected to
have seen statistics before, but as you will see,
you are expected to have seen probability. And usually, you do
see some statistics in a probability course. So I'm sure some of
you have some ideas, but I won't expect anything. And we'll be using mathematics. Math class, so there's going
to be a bunch of equations-- not so much real data
and statistical thinking. We're going to try to provide
theoretical guarantees. We have two estimators
that are available for me-- how theory guides me to choose
between the best of them, how certain can I be of my
guarantees or prediction? It's one thing to
just bid out a number. It's another thing to put
some error bars around. And we'll see how to build
error bars, for example. You will have your
own applications. I'm happy to answer questions
about specific applications. But rather than trying
to tailor applications to an entire institute,
I think we're going to work with pretty
standard applications, mostly not very serious ones. And hopefully, you'll be able
to take the main principles back with you and apply them to
your particular problem. What I'm hoping that you will
get out of this class is that when you have a real-life
situation-- and by "real life", I mean mostly at MIT, so some
people probably would not call that real life-- their goal is to formulate
a statistical problem in mathematical terms. If I want to say,
is a drug effective, that's not in
mathematical terms, I have to find out
which measure I want to have to call it effective. Maybe it's over a
certain period of time. So there's a lot of things
that you actually need. And I'm not really
going to tell you how to go from the application
to the point you need to be. But I will certainly
describe to you what point you need to be at
if you want to start applying statistical methodology. Then once you understand
what kind of question you want to answer-- do I want a yes/no answer,
do I want a number, do I want error bars, do
I want to make predictions five years into future, do
I have side information, or do I not have side
information, all those things-- based on that,
hopefully, you will have a catalog of
statistical methods that you're going to be able to
use and apply it in the wild. And also, no statistical
method is perfect. Some of the math people have
agreed upon over the years, and people understand
that this is the standard. But I want you to be
able to understand what the limitations are,
and when you make conclusions based on data, that those
conclusions might be erroneous, for example. All right, more practically, my
goal here is to have you ready. So who has taken, for example,
a machine-learning class here? All right, so many of you,
actually-- maybe a third have taken a
machine-learning class. So statistics has somewhat
evolved into machine learning in recent years. And my goal is to
take you there. So machine learning has a
strong algorithmic component. So maybe some of you have
taken a machine-learning class that displays mostly the
algorithmic component. But there's also a
statistical component. The machine learns from data. So this is a statistical track. And there are some statistical
machine-learning classes that you can take here. They're offered at the
graduate level, I believe. But I want you to be ready to
be able to take those classes, having the statistical
fundamentals to understand what you're doing. And then you're going to be able
to expand to broader and more sophisticated methods. Lectures are here from 11:00 to
12:30 on Tuesday and Thursday. Victor-Emmanuel will also be-- and you can call him Victor-- will also be holding
mandatory recitation. So please go on Stellar
and pick your recitation. It's either 3:00 to 4:00 or
4:00 to 5:00 on Wednesdays. And it's going to be mostly
focused on problem-solving. They're mandatory in the sense
that we're allowed to do this, but they're not going to
cover entirely new material. But they might cover
some techniques that might save you some time
when it comes to the exam. So you might get by. Attendance is not going to be
taken or anything like this. But I highly
recommend that you go, because, well,
they're mandatory. So you cannot really complain
that something was taught only in recitation. So please register
on Stellar for which of the two recitations
you would like to be in. They're capped at 40, so
first come, first served. Homework will be due weekly. There's a total of
11 problem sets. I realize this is a lot. Hopefully, we'll
keep them light. I just want you to
not rush too much. The 10 best will
be kept, and this will count for a total of
30% of the final grade. There are due Mondays
at 8:00 PM on Stellar. And this is a new thing. We're not going to use the boxes
outside of the math department. We're going to use
only PDF files. Well, you're always welcome
to type them and practice your LaTeX or Word typing. I also understand that this
can be a bit of a strain, so just write them down
on a piece of paper, use your iPhone, and
take a picture of it. Dropbox has a nice, new-- so try to find something
that puts a lot of contrast, especially if you use
pencil, because we're going to check if they're readable. And this is your responsibility
to have a readable file. I've had over the years-- not at MIT, I must
admit-- but I've had students who actually
write the doc file and think that
converting it to a PDF consists in erasing
the extension doc and replacing it by PDF. This is not how it works. So I'm sure you
will figure it out. Please try to keep
them letter-sized. This is not a
strict requirement, but I don't want to
see thumbnails, either. You are allowed to have
two late homeworks. And by late, I
mean 24 hours late. No questions asked. You submit them,
this will be counted. You don't have to send
an email to warn us or anything like this. Beyond that, even that
you have one slack for one 0 grade and slack
for two late homeworks, you're going to have to come
up with a very good explanation why you need actually more
extensions than that, if you ever do. And particularly,
you're going to have to keep track about why you've
used your three options before. There's going to
be two midterms. One is October 3, and
one is November 7. They're both going to be
in class for the duration of the lecture. When I say they last for
an hour and 20 minutes, it does not mean that
if you arrive 10 minutes before the end of
lecture, you still get an hour and 20 minutes. It will end at the
end of lecture time. For this as well, no pressure. Only the best of the
two will be kept. And this grade will count
for 30% of the grade. This will be closed-books
and closed-notes. The purpose is for you to-- yes? AUDIENCE: How many midterms
did you say there are? PHILIPPE RIGOLLET: Two. AUDIENCE: You said the best
of the two will be kept? PHILIPPE RIGOLLET: I
said the best of the two will be kept, yes. AUDIENCE: So both the
midterms will be kept? PHILIPPE RIGOLLET: The best
of the two, not the best two. AUDIENCE: Oh. PHILIPPE RIGOLLET: We will add
them, multiply the number by 9, and that will be grade. No. I am trying to be nice, there's
just a limit to what I can do. All right, so the goal is
for you to learn things and to be familiar with them. In the final, you
will be allowed to have your notes with you. But the midterms are
also a way for you to develop some mechanism so
that you don't actually waste too much time on things that
you should be able to do without thinking too much. You will be allowed
to cheat sheet, because, well, you can
always forget something. And it will be
two-sided letters sheet, and you can practice yourself
as writing as small as you want. And you can put whatever you
want on this cheat sheet. All right, the final will
be decided by the register. It's going to be
three hours, and it's going to count for 40%. You cannot bring books, but
you can bring your notes. Yes. AUDIENCE: I noticed
that the midterm dates aren't dated in the syllabus. So I wanted to
make sure you know. PHILIPPE RIGOLLET: They are not? AUDIENCE: Yeah-- PHILIPPE RIGOLLET:
Oh, yeah, there's a "1" that's missing on
both of them, isn't there? Yeah, let's figure that out. The syllabus is the true one. The slides are so
that we can discuss, but the ones that's
on the syllabus are the ones that count. And I think they're also posted
on the calendar on Stellar as well. Any other question? OK, so the pre-reqs here-- and who has looked at the
first problem set already? OK, so those hands
that are raised realize that there is a true
prerequisite of probability for this class. It can be at the level
of 18.600 or 604.1. I should say "B" now. It's two classes. I will require you
to know some calculus and have some notions
of linear algebra, such as, what is a matrix,
what is a vector, how do you multiply those
things together, some notion of what
orthonormal vectors are. We'll talk about
eigenvectors and eigenvalues, but I remind you all of that. So this is not this
strict pre-req. But if you've taken
it, for example, it doesn't hurt to
go back to your notes when we get closer
to this chapter on principle-component analysis. The chapters, as they're listed
in the syllabus, are in order, so you will see when
it actually comes. There's no required textbook. And I know you tend
to not like that. You like to have your textbook
to know where you're going and what we're doing. I'm sorry, it's just this class. Either I would have to go
to a mathematical statistics textbook, which
is just too much, or to go to a more
engineering-type statistics class, which is just too little. So hopefully, the
problems will be enough for you to practice
the recitations. We'll have some problems
to solve as well. And the material will
be posted on the slides. So you should have
everything you need. There's plenty of
resources online if you want to expand
on a particular topic or read it as said
by somebody else. The book that I
recommend in the syllabus is this book called All of
Statistics by Wasserman. Mainly because of the
title, I'm guessing it has all of it in it. It's pretty broad. There's actually not that many. It's more of an
intro-grad level. But it's not very deep, but
you see a lot of the overview. Certainly, what
we're going to cover will be a subset
of what's in there. The slides will be
posted on Stellar before lectures before
we start a new chapter and after we're done with the
chapter, with the annotations, and also, with the typos
corrected, like for the exam. There will be some
video lectures. Again, the first one will be
posted on OCW from last year. But all of them will be
available on Stellar-- of course, module
technical problems. But this is an automated system. And hopefully, it will
work out well for us. So if you somehow have
to miss a lecture, you can always catch
it up by watching it. You can also play
at that speed 0.75 in case I end up
speaking too fast, but I think I've
managed myself so far-- so just last warning. All right, why should
you study statistics? Well, if you read the news, you
will see a lot of statistics. I mentioned machine learning. It's built on a
lot of statistics. If I were to teach this
class 10 years ago, I would have to explain to you
that data collection and making decisions based on data was
something that made sense. But now, it's
almost in our life. We're used to this idea that
data helps in making decisions. And people use data
to conduct studies. So here, I found a bunch
of press titles that-- I think the key word I was
looking for was "study finds"-- if I want to do this. So I actually did not bother
doing it again this year. This is all 2016, 2016, 2016. But the key word that I look
for is usually "study find"-- so a new study find-- traffic is bad for your health. So we had to wait for 2016
for data to tell us that. And there's a bunch of other
slightly more interesting ones. For example, one that you
might find interesting is that this study finds that
students benefit from waiting to declare a major. Now, there's a bunch
of press titles. There one in the MIT News
that finds brain connections, key to reading. And so here, we have an
idea of what happened there. Some data was collected. Some scientific
hypothesis was formulated. And then the data was here
to try to prove or disprove this scientific hypothesis. That's the usual
scientific process. And we need to understand how
the scientific process goes, because some of those things
might be actually questionable. Who is 100% sure that
study finds that students-- do you think that you
benefit from waiting to declare a major? Right I would be
skeptical about this. I would be like, I don't want
to wait to declare a major. So what kind of
thing can we bring? Well maybe this
study studied people that were different from me. Or maybe the study
finds that this is beneficial for a
majority of people. I'm not a majority. I'm just one person. There's a bunch
of things that we need to understand what
those things actually mean. And we'll see that
those are actually not statements about individuals. They're not even statements
about the cohort of people they've actually looked at. They're statements
about a parameter of a distribution
that was used to model the benefit of waiting. So there's a lot of questions. And there are a lot of
layers that come into this. And we're going to want to
understand what was going on in there and try to peel it off
and understand what assumptions have been put in there. Even though it looks like
a totally legit study, out of those studies,
statistically, I think there's going to be
one that's going to be wrong. Well, maybe not one. But if I put a
long list of those, there would be a few that
would actually be wrong. If I put 20, there would
definitely be one that's wrong. So you have to see that. Every time you see 20 studies,
one is probably wrong. When there are studies
about drug effects, out of a list of 100,
one would be wrong. So we'll see what that means
and what I mean by that. Of course, not only studies
that make discoveries are actually making
the press titles. There's also the press
that talks about things that make no sense. I love this first experiment--
the salmon experiment. Actually, it was
a grad student who came to a neuroscience
poster session, pulled out this
poster, and explained the scientific experiment
that he was conducting, which consisted in taking a
previously frozen and thawed salmon, putting it
in an MRI, showing it pictures of violent images, and
recording its brain activity. And he was able to discover a
few voxels that were activated by those violent images. And can somebody tell
me what happened here? Was the salmon responding
to the violent activity? Basically, this is just
a statistical fluke. That's just randomness at play. There's so many voxels
that are recorded, and there's so
many fluctuations. There's always a
little bit of noise when you're in those
things, that some of them, just by chance, got lit up. And so we need to understand
how to correct for that. In this particular
instance, we need to have tools that tell us that,
well, finding three voxels that are activated for
that many voxels that you can find in
the salmon's brain is just too small of a number. Maybe we need to find a clump
of 20 of them, for example. All right, so
we're going to have mathematical tools that help us
find those particular numbers. I don't know if you ever
saw this one by John Oliver about phacking. Or actually, it said p-hacking. Basically, what John
Oliver is saying is actually a full-length-- like
there's long segments on this. And he was explaining how
there's a sociology question here about how there's a
huge incentive for scientists to publish results. You're not going to
say, you know what? This year, I found nothing. And so people are
trying to find things. And just by searching,
it's as if they were searching for all
the voxels in a brain until they find one that
was just lit up by chance. And so they just run
all these studies. And at some point, one will
be right just out of chance. And so we have to be very
careful about doing this. There's much more complicated
problems associated to what's called
p-hacking, which consists of violating the basic
assumptions, in particular, looking at the data,
and then formulating your scientific
assumption based on data, and then going back to it. Your idea doesn't work. Let's just formulate
another one. And if you are doing
this, all bets are off. The theory that we're
going to develop is actually for a very
clean use of data, which might be a little unpleasant. If you've had an army of
graduate students collecting genomic data for a
year, for example, maybe you don't
want to say, well, I had one hypothesis
that didn't work. Let's throw all the
data into the trash. And so we need to find
ways to be able to do this. And there's actually a
course been taught at BU. It's still in its early
stages, but something called "adaptive data
analysis" that will allow you to do these kind of things. Questions? OK, so of course,
statistics is not just for you to be
able to read the press. Statistics will probably
be used in whatever career path you choose for yourself. It started in the 10th century
in Netherlands for hydrology. Netherlands is basically
under water, under sea level. And so they wanted
to build some dikes. But once you're going
to build a dike, you want to make sure
that it's going to sustain some tides and some floods. And so in particular,
they wanted to build dikes that were high
enough, but not too high. You could always
say, well, I'm going to build a 500-meter dike,
and then I'm going to be safe. You want something
that's based on data. You want to make sure. And so in particular,
what did they do? Well, they collected
data for previous floods. And then they just
found a dike that was going to cover
all these things. Now, if you look at the
data they probably had, maybe it was scarce. Maybe they had 10 data points. And so for those
data points, then maybe they wanted to
sort of interpolate between those points, maybe
extrapolate for the larger one. Based on what they've
seen, maybe they have chances of
seeing something which is even larger than everything
they've seen before. And that's exactly the goal
of statistical modeling-- being able to extrapolate
beyond the data that you have, guessing what you have
not seen yet might happen. When you buy insurance
for your car, or your apartment,
or your phone, there is a premium
that you have to pay. And this premium
has been determined based on how much you
are, in expectation, going to cost the insurance. It says, OK, this person
has, day a 10% chance of breaking their iPhone. An iPhone costs
that much to repair, so I'm going to
charge them that much. And then I'm going to add
an extra dollar for my time. That's basically how those
things are determined. And so this is using statistics. This is basically where
statistics is probably mostly used. I was personally
trained as an actuary. And that's me being a
statistician at an insurance company. Clinical trials-- this is also
one of the earliest success stories of statistics. It's actually now widespread. Every time a new drug is
approved for market by the FDA, it requires a very strict
regimen of testing with data, and control group,
and treatment group, and how many people
you need in there, and what kind of significance
you need for those things. In particular, those
things look like this, so now it's 5,000 patients. It depends on what
kind of drug it is, but for, say, 100
patients, 56 were cured, and 44 showed no improvement. Does the FDA consider that
this is a good number? Do they have a table for how
many patients were cured? Is there a placebo effect? Do I need a control
group of people that are actually getting a placebo? It's not clear,
all these things. And so there's a lot of
things to put into place. And there's a lot of
floating parameters. So hopefully, we're
going to be able to use statistical modeling
to shrink it down to a small number of
parameters to be able to ask very simple questions. "Is a drug effective" is
not a mathematical equation. But "Is p larger than 0.5?" is a mathematical
question And that's essentially we're
going to be doing. We're going to take this, is a
drug effective, to reducing to, is a variable larger than 0.5? Now, of course genetics
are using that. That's typically actually
the same size of data that you would
see for FMRI data. So this is actually
a study that I found. You have about 4,000 cases of
Alzheimer's and 8,000 control. So people without Alzheimer's--
that's what's called a control. That's something
just to make sure that you can see the
difference with people that are not affected by
either a drug or a disease. Is the gene APOE associated
with Alzheimer's disease? Everybody can see why this
would be an important question. We now have it crisper. It's targeted to
very specific genes. If we could edit it, or
knock it down, or knock it up, or boost it, maybe
we could actually have an impact on that. So those are very
important questions, because we have the technology
to target those things. But we need the answers
about what those things are. And there's a bunch
of other questions. The minute you're going to
talk to biologists about say, I can do that. They're going to
say, OK, are there any other genes
within the genes, or any particular snips
that I can actually look at? And they're looking at
very different questions. And when you start asking
all these questions, you have to be careful, because
you're reusing your data again. And it might lead you
to wrong conclusions. And those are all over
the place, those things. And that's why they go all
the way to John Oliver talking about them. Any questions about
those examples? So this is really a motivation. Again, we're not
going to just take this data set of those cases
and look at them in detail. So what is common to
all these examples? Like, why do we have
to use statistics for all those things? Well, there's the
randomness of the data. There's some effect that
we just don't understand-- for example, the randomness
associated with the lining up of some voxels. Or the fact that as
far as the insurance is concerned whether you're
going to break your iPhone or not is essentially
a coin toss. Fully, it's biased. But it's a coin toss. From the perspective
of the statistician, those things are
actually random events. And we need to tame
this randomness, to understand this randomness. Is this going to be
a lot of randomness? Or is it going to be
a little randomness? Is it going to be
something that's like, out of their people-- let's see, for example,
for the floods. Were the floods that I
saw consistently almost the same size? It was almost a rounding
error, or they're just really widespread. All these things, we
need to understand so we can understand
how to build those dikes or how to make decisions
based on those data. And we need to understand
this randomness. OK, so the associated
questions to randomness were actually
hidden in the text. So we talked about
the notion of average. Right, so as far as the
insurance is concerned, they want to know in average
with the probability is. Like, what is your chance of
actually breaking your iPhone? And that's what came in
this notion of fair premium. There's this notion
of quantifying chance. We don't want to talk
maybe only about average, maybe you want to cover say
99% percent of the floods. So we need to know what is
the height of a flood that's higher than 99% of the floods. But maybe there's 1%
of them, you know. When doomsday comes,
doomsday comes. Right, we're not
going to pay for it. All right, so that's
most of the floods. And then there's questions
of significance, right? So you know I give this
example, a second ago about clinical trials. I give you some numbers. Clearly the drug cured more
people than it did not. But does it mean that
it's significantly good, or was this just by chance. Maybe it's just that these
people just recovered. It's like you know
curing a common cold. And you feel like,
oh I got cured. But it's really you waited five
days and then you got cured. All right, so there's this
notion of significance, of variability. All these things
are actually notions that describe randomness
and quantify randomness into simple things. Randomness is a very
complicated beast. But we can summarize it into
things that we understand. Just like I am a
complicated object. I'm made of molecules,
and made of genes, and made of very
complicated things. But I can be summarized as
my name, my email address, my height and my weight,
and maybe for most of you, this is basically enough. You will recognize
me without having to do a biopsy on me
every time you see me. All right, so, to
understand randomness you have to go
through probability. Probability is the
study of randomness. That's what it is. That's what the first sentence
that a lecturer in probability will say. And so that's why I need the
pre-requisite, because this is what we're going to use
to describe the randomness. We'll see in a second how it
interacts with statistics. So sometimes, and actually
probably most of the time throughout your
semester on probability, randomness was very
well understood. When you saw a
probability problem, here was the chance of
this happening, here was the chance
of that happening. Maybe you had more
complicated questions that you had some basic
elements to answer. For example, the probability
that I have HBO is this much. And the probability that I watch
Game of Thrones is that much. And given that I play basketball
what is the probability-- you had all these
crazy questions, but you were able to build them. But all the basic numbers
were given to you. Statistics will be about
finding those basic numbers. All right so some examples
that you've probably seen were dice, cards,
roulette, flipping coins. All of these things
are things that you've seen in a probability class. And the reason is
because it's very easy to describe the probability
of each outcome. For a die we know that
each face is going to come with probably 1/6. Now I'm not going to go into
a debate of whether this is pure randomness or
this is determinism. I think as a model
for actual randomness a die is a pretty good
number, flipping a coin is a pretty good model. So those are actually
a good thing. So the questions that you
would see, for example, in probabilities
are the following. I roll one die. Alice gets $1 if the number
of dots is less than three. Bob gets $2 if the number
of dots is less than two. Do you want to be Alice or
Bob given that your role is actually to make money. Yeah, you want to be Bob, right? So let's see why. So if you look at the
expectation of what Alice makes. So let's call it a. This is $1, with
probability 1/2. So 3/6, that's 1/2. And the expectation
of what Bob makes, this is $2 with probably
2/6 and that's 2/3. Which is definitely
larger than 1/2. So Bob's expectations
actually a bit higher. So those are the kind
of questions that you may ask with probability. I described to you
exactly, you use the fact that the die would get
less than three dots, with probability one half. We knew that. And I didn't have to describe
to you what was going on there. You didn't have to
collect data about a die. Same thing, you roll two dice. You choose a number
between 2 and 12 and you win $100 if you choose
the sum of the two dice. Which number do you pick? What? AUDIENCE: 7. PHILIPPE RIGOLLET: 7. Why 7? AUDIENCE: It's the most likely. PHILIPPE RIGOLLET: That's
the most likely one, right? So your gain here will be
$100 times the probability that the sum of the two
dice, let's say x plus y, is equal to your little
z where a little z is the number you pick. So 7 is the most
likely to happen and that's the one that
maximizes this function of z. And for this you need to study
a more complicated function. But it's a function
that enables two die. But you can compute the
probability that x plus y is equal to z, for every
z between 2 and 12. So you know exactly what
the probabilities are and that's how you
start probability. So here that's
exactly what I said. You have a very simple process
that describes basic events. Probability 1/6
for each of them. And then you can
build up on that, and understand probably of
more complicated events. You can throw some
money in there. You can be build functions. You can do very complicated
things building on that. Now if I was a
statistician, a statistician would be the guy who
just arrived on earth, had never seen a die
and needs to understand that a die come up with
probably 1/6 on each side. And the way he would do
it is just to roll the die until he get some counts
and tries to estimate those. And maybe that guy
would come and say, well, you know,
actually, the probability that I get a 1 is 1/6 plus
0.001 and the probability that I get a 2 is
1/6 minus 0.005. And there would be some
fluctuations around this. And it's going to be his
role as a statistician to say, listen, this
is too complicated of a model for this thing. And these should all
be the same numbers. Just looking at data, they
should be all the same numbers. And that's part of the modeling. You make some
simplifying assumptions that essentially make your
questions more accurate. Now, of course, if
your model is wrong, if it's not true that
all the faces arrive with the same probability, then
you have a model error here. So we will be
making model errors. But that's going to
be the price to pay to be able to extract
anything from our data. So for more
complicated processes, so of course nobody's going to
waste their time rolling dice. I mean, I'm sure
you might have done this in AP stat or something. But the need is to estimate
parameters from data. All right, so for more
complicated things you might want to estimate
some density parameter on a particular set of material. And for this maybe you need
to beam something to it, and measure how fast
it's coming back. And you're going to have
some measurement errors. And maybe you need to
do that several times and you have a model for
the physical process that's actually going on. And physics is usually
a very good way to get models for
engineering perspective. But there's models
for sociology where we have no physical system, right. God knows how people interact. And maybe I'm going
to say that the way I make friends is by first
flipping a coin in my pocket. And with probability
2/3, I'm going to make my friend at work. And with probability
1/3 I'm going to make my friend at soccer. And once I make my
friends at soccer-- I decide to make
my friend soccer. Then I will face
someone who's flipping the same coin with maybe be
slightly different parameters. But those things actually exist. There's models about how
friendships are formed. And the one I
described is called the mixed-membership model. So those are models that
are sort of hypothesized. And they're more reasonable
than taking into account all the things that made
you meet that person at that particular time. So the goal here--
so based on data now, once we have the model is going
to be reduced to maybe two, three, four
parameters, depending on how complex the model is. And then your goal will be
to estimate those parameters. So sometimes the randomness
we have here is real. So there's some true
randomness in some surveys. If I pick a random
student, as long as I believe that my random
number generator that will pick your random
ID is actually random, there is something
random about you. The student that
I pick at random will be a random student. The person that I call on
the phone is a random person. So there's some randomness
that I can build into my system by drawing something from
a random number generator. A biased coin is a random thing. It's not a very
interesting random thing. But it is a random thing. Again, if I wash out the
fact that it actually is a deterministic mechanism. But at a certain accuracy,
a certain granularity, this can be thought of as
a truly random experiment. Measurement error for example,
if you by some measurement device. or some optics
device, for example. You will have like standard
deviation and things that come on the side of the box. And it tells you, this will be
making some measurement error. And it's usually thermal noise
maybe, or things like this. And those are very
accurately described by some random phenomenon. But sometimes, and I'd say most
times, there's no randomness. There's no randomness. It's not like you breaking
your iPhone is a random event. This is just something
that we sweep-- randomness is a big rug
under which we sweep everything we don't understand. And we just hope
that in average we've captured, the average
effect of what's going on. And the rest of it might
fluctuate to the right, might fluctuate to the left. But what remains is
just sort of randomness that can be averaged out. So, of course, this is
where the leap of faith is. We do not know whether we
were correct of doing this. Maybe we make some
huge systematic biases by doing this. Maybe we forget a very
important component. Right, for example, if I have-- I don't know, let's
think of something-- a drug for breast cancer. All right, and I
throw out the fact that my patient is
either a man or woman. I'm going to have some
serious model biases. Right. So if I say I'm going to
collect a random and patient. And said I'm going
to start doing this. There's some information
that I really need, clearly, to build into my model. And so the model should be
complicated enough, but not too complicated. Right so it should take
into account things there will systematically
be important. So, in particular, the
simple rule of thumb is, when you have a
complicated process, you can think of it as
being a simple process and some random noise. Now, again, the random
noise is everything you don't understand about
the complicated process. And the simple process is
everything you actually do. So good modeling,
and this is not where we'll be
seeing in this class, consistent choosing
plausible simple models. And this requires a tremendous
amount of domain knowledge. And that's why we're not
doing it in this class. This is not something where I
can make a blanket statement about making good modeling. You need to know, if I
were a statistician working on a study, I would have to
grill the person in front of me, the expert, for two hours
to know, but how about this? How about that? How does this work? So it requires to
understand a lot of things. There's this famous statistician
to whom this sentence is attributed, and it's
probably not his then, but Tukey said that he
loves being a statistician, because you get to play
in everybody's backyard. Right, so you get to
go and see people. And you get to understand, at
least to a certain extent, what their problems are. Enough that you
can actually build a reasonable model for what
they're actually doing. So you get to do some sociology. You get to do some biology. You get to do some engineering. And you get to do a lot
of different things. Right, so he was
actually at some point predicting the
presidential election. So, you see, you get to do
a lot of different things. But it requires a lot
of time to understand what problem you're working on. And if you have a particular
application in mind you're the best person to
actually understand this. So I'm just going to
give you the basic tools. So this is the circle of trust. No, this is really
just a simple graphic that tells you what's going on. When you do probability,
you're given the truth. Somebody tells you what
die God is rolling. So you know exactly what the
parameters of the problems are. And what you're trying
to do is to describe what the outcomes are going to be. You can say, if you're
rolling a fair die, you're going to have 1/6
of the time in your data you're going to have one. 1/6 of the time you're
going to have to have two. And so you can describe-- if
I told you what the truth is, you could actually
go into a computer, either generate some data. Or you could describe to me
some more macro properties of what the data would be like. Oh, I would see a
bunch of numbers that would be centered
around 35, if I drew from a Gaussian
distribution centered at 35. Right, you would know
this kind of thing. I would know that it's very
unlikely that if my Gaussian has standard deviation-- is centered on 0, say,
with standard deviation 3. It's very unlikely that I will
see numbers below minus 10 in above 10, right? You know this, that you
basically will not see them. So you know from the truth,
from the distribution of a random variable that does
not have mu or sigmas, really numbers there. You know what data,
you're going to be having. Statistics is about
going backwards. It's saying, if I have
some data, what was the truth that generated it. And since there are so
many possible truths, Modeling says you
have to pick one of the simpler possible truths,
so that you can average out. Statistics basically
means averaging. You're averaging when
you do statistics. And averaging
means that if I say that I received--
so if I collect all your GPAs, for example. And my model is that
the possible GPAs are any possible numbers. And anybody can have
any possible GPA. This is going to be
a serious problem. But if I can summarize
those GPAs into two numbers, say, mean and
standard deviation, than I have a pretty
good description of what is going on, rather
than having to have to predict the full list. Right, if I learn a full
list of GPAs and I say, well this was the distribution. Then it's not going to be of
any use for me to predict what the GPA would be, or some
random student walking in, or something like this. So just to finish my rant about
probability versus statistics, this is a question you would
see in a probability-- this is a probabilistic question, and
this is a statistical question. The probabilistic question
is, previous studies showed that the drug
was 80% effective. So you know that. This is the effectiveness
of the drug. It's given to you. This is how your problem starts. Then we can anticipate that,
for a study on 100 patients, in average, 80 be cured. And at least 65 will be
cured with 99% chances. So again these are not-- I'm not predicting on 100
patients exactly the number of them they're
going to be cured. And the number of
them that are not. But I'm actually
sort of predicting what things are going
to look like on average, or some macro properties of what
my data sets will look like. So with 99 percent
chances, that means that in 99.99% of the
data sets you will draw from this particular draw. 99.99% of the cohort of 100
patients to whom you administer this drug, I will be able
to conclude that at least 65 of them will be cured, on 99.99%
percent of those data sets. So that's a pretty
accurate prediction of what's going to happen. Statistics is the opposite. It says, well, I just know
that 78 out of 100 were cured. I have only one data set. I cannot make predictions
for all data sets. But I can go back
to the probability, make some inference about
what my probability will look like, and then say, OK, then
I can make those predictions later on. So when I start with
78/100 then maybe I'm actually, in this
case, I just don't know. My best guess here is
that I'm confident I have to add the extra error that
I bet you making by predicting that here, the drug is not 80%
effective but 78% effective. And they need some
error bars around this, that will hopefully contain 80%,
and then based on those error bars I'm going to make slightly
less precise predictions for the future. So, to conclude, so this
was, why statistics? So what is this course about? It's about understanding
the mathematics behind statistical methods. It's more of a tool. We're not going to have fun and
talk about algebraic geometry just for fun in
the middle of it. So it justifies quantitative
statements given some modeling assumptions, that we
will, in this class, mostly admit that the modeling
assumptions are correct. | the first part--
in this introduction, we will go through
them because it's very easy to forget what
the assumptions are actually making. But this will be a
pretty standard thing. The words you will
hear a lot are IID-- independent and
identically distributed-- that means that your data
is basically all the sams. And one data point is not
impacting another data point. Hopefully we can describe
some interesting mathematics arising in statistics. You know, if you've
taken linear algebra, maybe we can explain to you why. If you've done some
calculus, maybe we can do some
interesting calculus. We'll see how in the
spirit of applied math those things answer
interesting questions. And basically we'll try to
carve out a math toolbox that's useful for us statistics. And maybe you can extend it
to more sophisticated methods that we did not
cover in this class. In particular in the
immersion learning class, hopefully you'll be able to
have some statistical intuition about what is going on. So what this course
is not about, it's not about spending a lot
of time looking at data sets, and trying to understand
some statistical thinking kind of questions. So this is more of an applied
statistical perspective on things, or more modeling. So I'm going to typically
give you the model. And say this is a model. And this is how we're
going to build an estimator in the framework of this model. So for example, 18.075,
to a certain extent, is called "Statistical
Thinking and Data Analysis." So I'm hoping there is some
statistical thinking in there. We will not talk about
software implementation. Unfortunately, there's just
too little time in a semester. There's other courses that
are giving you some overview. So the main software
these days are R is the leading software I'd say
in statistics, both in academia and industry, lots of
packages, one every day that's probably coming out. But there's other things,
right, so now Python is probably catching up with all these
scikit-learn packages that are coming up. Julia has some
statistics in there, but it really if you were to
learn a statistical software, let's say you love
doing this, this would be the one that would
prove most useful for you in the future. It does not scale super well
to high dimensional data. So there is a class
an IDSS that actually uses R. It's called
IDS 0.12, I think it's called "Statistics,
Computation, and Applications," or something like this. I'm also preparing,
with Peter Kempthorne, a course called
"Computational Statistics." It's going to be offered this
Spring as a special topics. And so Peter Kempthorne
will be teaching it. And this class
will actually focus on using R. And
even beyond that, it's not just going
to be about using. It's going to be
about understanding-- just the same way we
we're going to see how math helps
you do statistics, it's going to help
see how math helps you do algorithims for statistics. All right, so we'll talk about
maximum likelihood estimator. Will need to maximize
some function. There's an optimization
toolbox to do that. And we'll see how we
can have specialized for statistics
for that, and what are the principles behind it. And you know, of
course, if you've taken AP stats you
probably think that stats is boring to death
because it was just a long laundry-list that
spent a lot of time on t-test. I'm pretty sure we're not going
to talk about t-test, well, maybe once. But this is not a matter of
saying you're going to do this. And this is a slight
variant of it. We're going to really try to
understand what's going on. So, admittedly, you have
not chosen the simplest way to get an A in
statistics on campus. All right, this is
not the easiest class. It might be
challenging at times, but I can promise you that
you will maybe suffer. But you will learn
something by the time you're out of this class. This will not be a
waste of your time. And you will be
able to understand, and not having to remember by
heart how those things actually work. Are there any questions? Anybody want to go to other
stats class on campus? Maybe it's not too late. OK. So let's do some statistics. So I see the time
now and it's 11:56, so we have another 30 minutes. I will typically
give you a three, four minute break if
you want to stretch, if you want to run
to the bathroom, if you want to check
your texts or Instagram. There was very little
content in this class, hopefully it was
entertaining enough that you don't need the break. But just in the future, so you
know you will have a break. So statistics, this is how it
starts, I'm French, what can I say I need to put
some French words. So this is not how office
hours are going to go down. Anybody know this sculpture
by a Rodin, The Kiss. Maybe probably The
Thinker is more famous. But this is actually
a pretty famous one. But is it really this
one, or is it this one. Anybody knows which one it is? This one? Or this one? AUDIENCE: The previous. PHILIPPE RIGOLLET: What's that? AUDIENCE: This one. PHILIPPE RIGOLLET:
It's this one. AUDIENCE: Final answer. PHILIPPE RIGOLLET: Yeah,
who votes for this one. OK. Who votes for that one? Thank you. I love that you do not want to
pronounce yourself with no data actually to make any decision. This is a total coin toss right. Turns out that there
is data, and there is in the very serious
journal Nature, someone published a
very serious paper which actually looks pretty serious. If you look at it, it's
like "Human Behavior: Adult persistence of
head-turning symmetry," is a lot of fancy
words in there. And this, I'm not
kidding you, this study is about collecting
data of people kissing, and knowing if they bend
their head to the right or if they bend they
head to the left. And that's all it is. And so a neonatal
right-side preference makes a surprising romantic
reappearance in later life. There's an explanation for it. All right, so if we follow this
Nature which one is the one. This one? Or this one? This one, right? Head to the right. And to be fair, for
this class I was like, oh, I'm going to go and show
them what Google Images does. When you Google
kissing couple, it's inappropriate after
maybe the first picture. And so I cannot show you this. But you know you can
check for yourself. Though I would
argue, so this person here actually went
out in airports and took pictures of strangers
kissing and collecting data. And can somebody guess why
did he just not stay home and collect data
from Google Images by just googling
kissing couples. What's wrong with this data? I didn't know actually before I
actually went on Google Images. AUDIENCE: It can be altered? PHILIPPE RIGOLLET:
What was that? AUDIENCE: It can be altered. PHILIPPE RIGOLLET:
It can be altered. But, you know, who
would want to do this? I mean there's no
particular reason why you would want to flip an image
before putting it out there. I mean, you might, but
you know maybe they want to hide the brand of
your Gap shirt or something. AUDIENCE: I guess the people
who post pictures of themselves kissing on Google Images
are not representative of the general population. PHILIPPE RIGOLLET:
Yeah, that's very true. And actually it's
even worse than that. The people who post
pictures of themselves, are not posting
pictures of themselves or putting pictures
of the people that they took a picture of. And there usually is a
stock watermark on this. And it's basically stock images. Those are actors, and so
they've been directed to kiss and this is not a
natural thing to do. And actually, if you go
to Google Images-- and I encourage you to
do this, unless you don't want to see
inappropriate pictures, and they're mightily
inappropriate. And basically you will see
that this study is actually not working at all. I mean, I looked briefly. I didn't actually
collect numbers. But I didn't find a particular
tendency to bend right. If anything, it was actually
probably the opposite. And it's because those people
were directed to do it. They just don't actually
think about doing it. And also because
I think you need to justify writing in
your paper more than, I sat in front of my computer. So again, this
first sentence here, a neonatal right-side
preference-- "is there a right
side preference?" is not a mathematical question. But we can start saying, let's
blah, and put some variables, and ask questions
about those variables. So you know x is actually
not a variable that's used very much in
statistics for parameters. But p is one, for parameter. And so you're going to take
your parameter of interest, p, As here is going to be
the proportion of couples. And that's among all couples. So here, if you talk about
statistical thinking, there would be a question about
what population this would actually be representative of. | usually this is
a call to your-- sorry, I should not forget this
word it's important for you. OK, I forget this word. So this is-- OK, So if you look at
this proportion, maybe these couples
that are in the study might be representative
only of couples in airports. Maybe they actually put on a
show for the other passengers. Who knows? You know, like, oh,
let's just do it as well. And just like the
people in Google Images they are actually doing it. So maybe you want
to just restrict it. But of course clearly if
it's appearing in Nature, it should not be only
about couples in airports. It's supposedly representative
of all couples in the world. And so here let's
just keep it vague, but you need to keep
in mind what population this is actually making
a statement about. So you have this full population
of people in the world. Right, so those are
all the couples. And this person went
ahead and collected data about a bunch of them. And we know that, in this
thing, there's basically a proportion of
them, that's like p, and that's the proportion
of them that's bending their head to the right. And so everybody on this side
is bending their heads right. And hopefully we
can actually sample this thing you're informing. That's basically the
process that's going on. So this is the
statistical experiment. We're going to observe
n kissing couples. So here we're going to put
as many variables as we can. So we don't have to
stick with numbers. And then we'll just
plug in the numbers. n kissing couples, and n
is also, in statistics, by the way, n is the size of
your sample 99.9% of the time. And collect the value
of each outcome. So we want numbers. We don't want right or left. So we're going to code them
by 0 and 1, pretty naturally. And then we're going to
estimate p which is unknown. So p is this area. And we're going to
estimate it simply by the proportion of right
So the proportion of crosses that actually fell
in the right side. So in this study
what you will find is that the numbers
that were collected were 124 couples, and that,
out of those 124, 80 of them turned their head to the right. So, p hat is a proportion. How do we do it? Well, you don't need
statistics for that. You're going to see
80 divided by 124. And you will find that
in this particular study 64.5% of the
couples were bending their heads to the right. That's a pretty
large number, right? The question is if I picked
another 124 couples, maybe at different airports, different
times, would I see same number? Would this number be
all over the place? Would it be sometimes very
close to 120, or sometimes for close to 10? Or would it be-- is this number
actually fluctuating a lot. And so, hopefully not too much,
64.5 percent is definitely much larger than 50%. And so there seems to
be this preference. Now we're going to
have to quantify how much of this preference. Is this number significantly
larger than 50%? So if our data, for example,
was just three couples. I'm just going there,
I'm going to Logan. I call it, I do
right, left right. And then I see-- see what's the name of
the fish place there? I go to I go to Wahlburgers
at Logan and I'm like, OK, I'm done for the day. I collect this data. I go home, and I'm like,
wow, 66.7% to the right. That's a pretty big number. It's even farther from
50% than this other guy. So I'm doing even better. But of course you know
that this is not true. Three people is definitely
not representative. If I stopped at the
first one, I would have actually-- at the first
two, I would have even 100%. So the question that statistics
is going to help us answer is, how large should the sample be? For some reason, I don't know
if you guys receive this, I'm an affiliate with
the Broad Institute, and since then I receive
one email per day that says, sample
size determination-- how large should your sample be? Like, I know how large
should with my sample be. I've taken 18.650
multiple times. And so I know, but the
question is-- is 124 a large enough number or not? Well, the answer is actually,
as usual, it depends. It will depend on the
true unknown value of p. But from those particular
values that we got, so 120 and-- how many couples was there? 80? We actually can
make some question. So here we said that
80 was larger than 50-- was allowing us to
conclude at 64.5%. So it could be one reason to
say that it was larger than 50%. 50% of 124 is 62. So the question is,
would I be would I be willing to make
this conclusion at 63? Is that a number that
would convince you? Who would be convinced by 63? who would be convinced by 72? Who would be convinced by 75? Hopefully the number of hands
that are raised should grow. Who would be convinced by 80? All right, so basically
those numbers actually don't come from anywhere. This 72 would be the number that
you would need for a study-- most statistical studies
would be the number that they would retain. That's not for 124. You would need to see
72 that turn their head right to actually
make this conclusion. And then 75-- So we'll see that there's many
ways to come to this conclusion because, as you
can see, this was published in Nature with 80. So that was OK. So 80 is actually a
very large number. This is 99 point-- this 99% -- no, so
this is 95% confidence. This is 99% confidence. And this is 99.9%
percent confidence. So if you said 80 you're a
very conservative person. Starting at 72, you can
start making this conclusion. To understand
this, we need to do our little mathematical
kitchen here, and we need to do some modeling. So we need to
understand by modeling-- we need understand what
random process we think this data is generating from. So it's going to have
some unknown parameters, unlike in probability. But we need to have just
basically everything written except for the values
of the parameters. When I said a die is coming
uniformly with probably 1/6 then I need to have, say
maybe with probability-- maybe I should say here
are six numbers, and I need to just
fill those numbers. So for i equal 1 to
n, I'm going to define Ri to be the indicator. An indicator is just something
that takes value 1 if something is true, and 0 if not. So it's an indicator
that i-th couple turns the head to the right. So, Ri, so it's indexed by i. And it's one if the i-th couple
turns their head to the right, and 0 if it's-- well actually, I guess they can
probably kiss straight, right? So that would be weird, but
they might be able to do this. So let's say not right. Then the estimator of
p, we said, was p hat. It was just the
ratio of two numbers. But really what it is is
I count, I sum those Ri's. Since I only add those that
take value 1, what this is is-- this sum here is actually just
counting the number of 1's. Which is another way to say it's
counting the number of couples that are kissing to the right. And here I don't even
have to tell you anything about the numbers or anything. I can only keep track of-- first couple is a 0
second couple is a 1, third couple is 0. The data set-- you can
actually find it online-- is actually a sequence
of 0's and 1's. Now clearly for the
question that we're asking about this
proportion, I don't need to keep track of
all this information. All I need to keep
track of is the number of 0's and the number of 1's. Those are completely
interchangeable. There's no time effect in this. The first couple is no
different than the 15th couple. So we call this Rn bar. That's going to be a very
standard notation that we use. R might be replaced by
other letters like x-- so xn bar, yn bar. And this thing
essentially means that I average the R's, or the
Ri's over n of them. And the bar means the average. So I divide by n the
total number of 1's. So here this sum was equal
to 80 in our example and n was equal to 124. Now this is an estimator. So an estimator is
different from an estimate. An estimate is a number. My estimate was 64.5. My estimator is this thing where
I keep all the variables free. And in particular, I
keep those variables to be random because I'm going
to think of a random couple kissing left to right as the
outcome of a random process, just like flipping a coin
be getting heads or tails. And so this thing here
is a random variable, Ri. And this average is, of course,
an average of random variables. It's itself a random variable. So an estimator is
a random variable. An estimate is the realization
of a random variable, or, in other words,
is the value that you get for this random variable
once you plug in the numbers that you've collected. So I can talk about the
accuracy of an estimator. Accuracy means what? Well, what would we
want for an estimator? Maybe we won't want it
to fluctuate too much. It's a random variable. So I'm talking about the
accuracy of a random variable. So maybe I don't want
it to be too volatile. I could have one
estimator which would be-- just throw out 182
couples, keep only 2 and average those two numbers. That's definitely
a worse estimator than keeping all of the 124. So I need to find
a way to say that. And what I'm going
to be able to say is that the number is
going to be fluctuating. If I take another
two couples, I'm going to be I'm
probably going to get a completely different number. But if I take another 124
couples two days later, maybe I'm going to have
a very number that's very close to 64.5%. So that's one way. The other thing we would like
about this estimator it's actually-- maybe it's not too
volatile-- but also we want it to be close to the
number that we're looking for. Here is an estimator. It's a beautiful variable. 72%, that's an estimator. Go out there just do
your favorite study about drug performance. And then they're going to
call you, MIT student taking statistics, they
say, so how are you going to build your estimator? We've collected those 5,000
or something like that. I'm just going to spit out 72%. Whatever the data says,
that's an estimator. It's a stupid estimator
but it is an estimator. But this is estimator
is very not volatile. Every time you're going
to have a new study, even if you change fields,
it's still going to be 72%. This is beautiful. And the problem is
that's probably not very close to the value you're
actually trying to estimate. So we need two things. We need are estimated
to be a random variable. So think in terms of densities. We want the density
to be pretty narrow. We want this thing
to have very little-- so this is definitely
better than this. But also, we want the number
that we're interested in, p, to be very close to this-- to be close to the values that
this thing is likely to take. If p is here, this is
not very good for us. So that's basically the things
we're going to be looking at. The first one is
referred to as variance. The second one is
referred to as bias. Those things come all
over in statistics. So we need to
understand a model. So here's the model that we have
for this particular problem. So we need to make assumptions
on the observations that we see. So we said we're going to assume
that the random variable-- that's not too much
of a leap of faith. We're just sweeping under
the rug everything thing we don't understand
about those couples. And the assumption
that we make is that Ri is a random variable. This one you will
forget very soon. The second one is that
each of the Ri's is-- so it's a random variable
that takes value 0 or 1. Anybody can suggest
the distribution for this random variable? AUDIENCE: Bernoulli. PHILIPPE RIGOLLET: What? AUDIENCE: Bernoulli. PHILIPPE RIGOLLET:
Bernoulli, right? And it's actually beautiful. This is where you have to do
the least statistical modeling. A random variable that
takes value 0 or 1 is always a Bernoulli. That's the simplest variable
you can ever think of. Any variable that takes
only two possible values can be reduced to a Bernoulli. OK, so this is a Bernoulli. And here we make the
assumption that it actually takes parameter p. And there's an assumption here. Anybody can tell me
what the assumption is? AUDIENCE: It's the same. PHILIPPE RIGOLLET:
Yeah, it's same, right? I could have said
p i, but it's p. And that's where I'm
going to be able to start getting to do some statistics. It's that I'm going to start
to be able to pull information across all my guys. If I assume that
they're all pi's completely uncoupled
with each other. Then I'm in trouble. There's nothing I
can actually get. And then I'm going to
assume that those guys are mutually independent. And most of the time they
will just say independent. Meaning that, it's not like all
these guys called each other and it's actually a flash mob. And they were like, let's all
turn our left side to the left. And then this is
definitely not going to give you a valid conclusion. So, again. randomness is
a way of modeling lack of information. Here there is a way
to figure it out. Maybe I could have
followed all those guys, and knew exactly what
they were-- maybe I could have looked at
pictures of them in the womb and guess how they were
turning-- by the way that's one of the conclusions,
they're guessing that we turn our
head to the right because our head is turned
to the right in the womb. So we don't know what goes
on in the kissers minds. And there's, you know,
physics, sociology. There's a lot of things
that could help us, but it's just too
complicated to keep track of, or too expensive
for many instances Now again, the nicest
part of this modeling was the fact that Ri's
take only two values, which mean that this conclusion
that they were Bernoulli was totally free for us. Once we know it's a random
variable, it's a Bernoulli. Now they could have
been, as we said, they could have been a
Bernoulli with parameter p i. For each i, I could have
put a different parameter, but I just don't have
enough information. What would I have said? I would say, well the first
couple turned to the right. p1 has to be one,
that's my best guess. The second couple
kiss to the left, well, p2 should be 0,
that's my best guess. And so basically I
need to have to be able to average my information. And the way I do it is by
coupling all these guys, pi's to be the same p for all i. OK, does it make sense? Here what I am assuming is that
my population is homogeneous. Maybe it's not. Maybe I could actually
look at a finer grain, but I'm basically making a
statement about a population. And so maybe you kiss to the
left, and then you're not-- I'm not making a statement
about a person individually, I'm making a statement about
the overall population. Now independence is
probably reasonable, right? This person just went
and know can seriously hope that these couples did not
communicate with each other. Or that you know Tanya did
not text that we should all turn our head to the left now. And there's no external
stimulus that forces people to do something different. OK, so-- sorry about that. Since we have about
less than 10 minutes. Let's do a little bit of
exercises, is that OK with you? So I just have some
exercises so we can see what an exercise going to look like. This is sort of similar to the
exercises you will see with me. We should do it together, OK? So now we're going to have-- I have a test. So that's an exam
in probability. OK. And I'm going to have 15
students in this test. And hopefully, this
should be 15 grades that are representative of the
grades of all a large class. Right, so if you go you know
18.600, it's a large class, there's definitely
more than 15 students. And maybe, just by sampling
15 students at random, I want to have an idea of what
my grade distribution will look like. I'm grading them, I want
to make an educated guess. So I'm going to make
some modeling assumptions for those guys. So here, 15 students and
the grades are x1 to x15. Just like we had R1,
R2, all the way to R124. Those were my Ri's. And so now I have my xi's. And I'm going to
assume that xi follows a Gaussian or normal
distribution with min mu and variance sigma squared. Now this is modeling, right? Nobody told me there's
no physical process that makes this happen. We know that there's something
called the central limit theorem in the background
that says that things tend to be Gaussian, but this is
really a matter of convenience. Actually this is, if
you think about it, this is terrible because this
puts non-zero probability on negative scores. I'm definitely not going
to get a negative score. But you know it's good
enough because they know the probabilities
non-zero but it's probably 10 to the minus 12. So I would be very unlucky
to see a negative score. So here's the list of grades, so
I have 65, 41, 70, 90, 58, 82, 76, 78-- maybe I should have done
it with 8 --59, 59-- sitting next to each other
--84, 89, 134, 51, and 72. So those are the
scores that I got. There were clearly some
bonus points over there. And the question is,
find estimator for mu. What is my estimator for mu? Well, an estimator,
again, is something that depends on the random variable. All right, so mu is
the expectation, right? So a good estimator is
definitely the average score, just like we had the
average of the Ri's. Now the xi's no longer need
to be 0's and 1's, so it's not going to boil down to being
a number of ones divided by the total numbers. Now if I'm looking
for an estimate, well, I need to actually
sum those numbers and divide them by 15. So my estimate is
going to be 1/15. Then I'm going to start
summing those numbers-- 65 plus 72. OK, and I can do
it, and it's 67.5. This is my estimate. Now if I want to compute
a standard deviation-- so let's say estimate for sigma. You've seen that before, right? An estimate for sigma is what? An estimate for sigma, we'll
see methods to do this, but sigma squared
is the variance, or is the expectation, of x
minus expectation of x squared. And the problem is
that I don't know what those expectations are. And so I'm going to do what
99.9% percent of statistics is. And what is statistics about? What's my motto? Statistics is about replacing
expectations with averages. That's what all of
statistics is about. There's 300 pages in a purple
book called All of Statistics that tells you this. All right, and then
you do something fancy. Maybe you minimize
something after you replace the expectation. Maybe you need to
plug in other stuff. But really, every time
you see an expectation, you replace it by an average. OK let's do this. So sigma squared
hat will be what? It's going to be 1 over n,
sum from i equals 1 to n of xi minus-- well, here I need to replace
my expectation by an average, which is really this average. I'm going to call
it mu hat squared. There, you have replaced my
expectation with average. OK so the golden thing
is, take your expectation and replace it with this. Frame it, get a tattoo, I don't
care but that's what it is. If you remember one thing from
this class, that's what it is. Now you can be fancy, if
you look at your calculator, it's going to put an n
minus 1 here because it wants to be unbiased. And those are things we
are going to come to. But let's say right
now we stick to this. And then when I
plug in my numbers. I'm going to get an
estimate for sigma, which is the square
root of the estimator once I plug in the numbers. And you can check that the
number, you get will be 18. So those are basic things and
if you've taken any AP stats this should be completely
standard to you. Now I have another list, and
I don't have time to see it. It doesn't really matter. OK, we'll do that next time. This is fine. We'll see another list
of numbers and see-- we're going to think about
modeling assumptions. The goal of this exercise is
not to compute those things, it's really to think about
modeling assumptions. Is it reasonable to think
that things are IID? Is it reasonable
to think that they have all the same parameters,
that they're independent, et cetera, OK so one thing that I wanted
to add is, probably by tonight, so I will try to use-- in the spirit of-- I don't know what's
starting to happen. In the spirit of using
my iPad and fancy things, I will try to post some
videos of-- for in particular, who has never used a
statistical table to read, say, the quantiles of a
Gaussian distribution? OK, so there's several of you. This is a simple
but boring exercise. I will just post a
video on how to do this, and you will be able
to find it on Stellar. It's going to take five
minutes, and then you will know everything there
is to know about those things but that's something you need
for the first problem set. By the way, so the
problem set has 30 exercises in probability. You need to do 15. And you only need to turn in 15. You can turn in all
of 30 if you want. But you need to know, by the
time we hit those things, you need to know-- well actually, by next week you
need to know what's in there. So if you don't have time
to do all the homework, and then go back to
your probability class to figure out how to do it,
just do 15 easy that you can do. And return those things. But go back to your
probability class and make sure that you
know how to do all of them. Those are pretty
basic questions, and those are things that I'm
not going to slow down on. So you need to remember that
the expectation of the product of independent
random variables is a product of the expectations. Expectation of the sum, is
the sum of the expectation. This kind of thing,
which is a little silly, but it just requires
you practice. So, just have fun. Those are simple exercises. You will have fun remembering
your probability class. All right, so I'll
see you on Tuesday-- or Monday.
