okay? So, that's the notion of a distribution. I'm now going to talk about

few characteristics of this distribution which may be very familiar for people who

are, have a statistical background but not familiar for all this, okay? So, let's

stick with our problem. And let's suppose, I know, the distribution of possible

heights. I want to calculate what is the normal. Imagine, if in my head, I had to

keep the heights of all people taking all classes in the university, it would be, it

would be mind boggling. So, what do we do? A distribution characterizes all

possibilities, but t hen I ask myself, what is the average chance? What is the

average height, sorry, right? And this, we call many times, expectation. And if the

distribution is normal, we can only worry about mean. It turns out the beauty of a

normal distribution is if I divide this, if I divide it over, if I carry this over,

it'll look like one perfect line, right? It's very symmetric. So, the mean is right

in the middle, and the min will also be equal to mode and median. I am getting a

little geeky now. These are two other ways of measuring what is called central

tendency. So, why am I interested in what's happening on average? Because

that's what most people think about the future. Hey, well, on average what will be

my cash flow next year? 100. But will it be exactly 100? No. It could be 90, it

could be 110. I hate the 90 but I like the 110. That's where the hypocrisy comes in,

okay? So, min, median and mode. I am sticking with heights. Let's figure out

how do we calculate that min, okay? And I've given you examples with returns and

so on, so, because we are doing finance, but I'm just getting a little excited

here. So, the way you'll figure out why, min, and you'll call it y bar. And

theoretically, it's also called expectation of y, will be equal to this.

What will you do? You'll take the values of y, all values of yi, all the values.

And you'll multiply them by Pi, which is what? The probability. So , you'll take

each Y, multiply it by its probability and sum, over how many? All n possible. So, if

i goes from one through n. N is the sample size okay? So, so, the, so what, what is

it saying? It's saying, multiply the probability by the chance, by the height.

And if the probability of being five feet seven, is one%, that's how you get the

first data point and so on. I want to just emphasize this way of doing things,

because I think people forget, that the usual way of, saying it, and I'm going to

write it up here, is this, summation yi / n. That's the usual way. You'll see it

done even in Excel. So, when you do min or average, i t's called in Excel, you tell

them what the ys are. They're already in a spreadsheet going from A1 through A100, if

there are 100 observations. You just sum them and divide by n. You're making an

assumption when you do that. And the assumption is, what is HPi? One . That

means that the chance of each height entered in your Excel spreadsheet, and by

the way, there's a note that tells you how to do that in Excel. It's so simple. You

just say, Excel says, do average. And then, we have the time towards the end, we

may do that. But I'm not inclined to do that right now. I just want you to

understand. It's very straightforward. Now, the assumption will tend on a normal

average that you calculate, right? So, what is the average rainfall this year?

What will they do on the website, on a weather website? They'll add up all the

rainfall for each day and divide by 365. They're assuming that the likelihood of

each thing is equal. And that's an important assumption. If you have a large

data set, it usually is an okay assumption, right? Because, it doesn't

matter what value of one / n is that much. I want to emphasize this so that you

understand. So, you've calculated. Okay, what do I expect will happen? However,

that's not the only story. I also have to worry about uncertainty or variance,

right? In this case, worrying about the variance of height is, doesn't seem that

traumatic but let's just stick with it, just as an example. Worrying about

variance of returns is very traumatic, right? Especially, if they're going in a

negative direction. So, this is what you have. So, what have I calculated? Let's

assume, I've already calculated y-bar. Remember, probabilities are here, Ps. Now,

what am I, I look at this and I say, ask you the following. Okay, are you sure? And

suppose the average height, in all the classes I've ever taught is five foot,

eight inches. And I, somebody asked me, but Gautam, are you sure that's the

height? I said, obviously I am not sure, right? The only way I'd be sure is this

height was how much? Exactly one. Then you wouldn't have a distribution, right? So,

you, are you sure? And the answer is, obviously, you're not. Some people are

here, some people are here. So, what do you do? You do this. You take a yi, each

yi. And suppose that's this one. And you subtract y-bar from it. Why? Because y-bar

is the normal, the center of gravity of this behavior, the normal behavior. So,

you got a deviation from it. In this, case it's positive, in this case, it's

negative, right? Now, you have to multiply that by the chance of this happening,

right? This data point happening, okay? But you have to do another thing, you have

to square it. And then, you sum across all possibilities. And that's called variance.

And the symbol used is sigma i^2. Quick question, think about it for a second. Why

do I not sum these? Why do I square them? And the reason is, I just gave you a hint.

The min is the center of gravity. So, what will happen? The positives and the

negatives will cancel each other out. And what will you get every time? Zero. So,

you, there's no point saying zero variance, because zero variance is only

true for what? Something is, you're a 100% sure about. Everybody's five feet, eight

inches tall or I'm going to get my money for sure. So, the variance is a measure of

uncertainty. However, look at its units. The units of average are what? Inches. The

unit of variance or uncertainty about your estimate or average height is squared. So,

what do we do? To make it the same unit, we do square root of sigma square i, which

we call sigma i, which we call standard deviation. By the way, one thing very

important to note about normal distributions is, just like the min is the

average, is also the median, is also the mode. Similarly, the only measure of

uncertainty is standard deviation. If you do more strange distributions, you'll get

things like skewness, kurtosis. I don't want to get into those, because that's not

the purpose of this class. High level possibilities of including skewness and

I'll be doing Finance. It depends o n your a ssumption about the distribution. But

for now, let's stick with standard deviation, okay? I'm going to keep going

and I'm going to first emphasize now why will we not stop here. Think about it.

Normal distribution, you know, the expected value, you know the uncertainty,

why? We are done with it. We know the measure of risk, right? Because we know

variance will be zero in the cash flows of each instrument if you're holding a

Treasury bill. But if you're holding a corporate bond, what will the variance be?

Positive, right? So, why worry about anything more? Why not just simply state

with not knowing the world and characterizing expectation by min, or

average, and uncertainty by variance? Well, there's a reason for it. And I'm

going to just give you a flavor of the reason before I do the Statistics because

we're going to get into the details of this concept, big time, next week when we

talk about measurement of risk. Why variance of a security versus portfolio?

Turns out, because we are risk averse, we are averse to risk. We hold portfolios. In

fact, I don't know anybody in the world who has money to invest who doesn't hold

portfolios. It will be silly to put all your eggs in one basket assuming you're

risk averse, and human behavior is risk averse. If there's enough data to show it,

and I'll show you more as we go along, including today. Because we hold port,

portfolios, portfolios are a collection of things. They are not single things. So,

not, so, imagine a world in which each one of us was holding just one thing. Either

Apple, Google, GE, and so on, and that was our behavior. That's not what the world is

like. Then, variances and mins would be enough. Turns out, I know ahead of time,

in fact, we knew it in the cave, when she was raised to live hunting outside for the

first time, guess what the guy said? Hey, don't put all your eggs in one basket.

That means, diversify, try to do different things so that you have different ways of

collecting food, so that you survive, right? So, risk aversion implies hol ding

po rtfolios. Portfolios means a collection of things, not single things. And that

means. Relations. We have to figure out relationships and how to measure them. Let

me ask you this simple example. I know I use very bizarre examples. Again, not to

do with Finance. Suppose, a human being could survive in, by themselves, just by

themselves, each single person, nothing to do with anybody else. Well, that's one

world. But what happens? We believe that, especially in business schools, we teach

group work. So, imagine, if you will, the only person in a, in a group, you're only

one thing, right? Now, let me ask you. If you have a collection of things, it's

called team. So, think about it. I could look at your behavior alone if you were

the only thing determining everything, right? You operate individually. But if

you operate in groups, and lets take a group of two, what have I done? How many

personalities? Two personalities. But what else have I introduced? I have introduced

relationships. How many? Me and Ryan, Ryan and me are a team doing this. What is

important now? Not just his personality and my personality. What's important is my

relationship with him, and his relationship with me. So, as soon as

collective, things in a portfolio or collections matter, we've got to be able

to measure relationships. And that's what, after a break, we'll try to do, using

Statistics. So please take a break, and we'll come back to how do you measure