WEBVTT

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Please.

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Okay.

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Okay.

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Hello.

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Hi.

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Hello.

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Okay.

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Okay, so.

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Things like they started working this would be probability.

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Okay Thursday. Okay. And I'm trying to record that and so on.

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So 1st, um.

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To remind you for the exam 1.

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It's in class that would be here, um, and and it's going to use.

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Um, he was great scope so, Frank.

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You know, bring your laptops or something, and you may be, you know, bring a scratch paper. If you want.

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You know, you may want to the question might ask you to compute, you know, work out something and then pick an answer off of grade school.

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Okay, um, it's a material.

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Up to the last Thursday, so.

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Class, so, perhaps, oh, you can also bring.

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Um, during 1 cheat sheet.

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This is all written down. Okay. Um.

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For office hours.

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Um, well, there's 1 on Friday, there's 1 on Saturday.

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And is going to be 1 on Sunday also, so that people want to ask questions.

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All on Webex so, of course, the 1 on Friday and, um.

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So, Friday at 3 o'clock, uh, Saturday at 3 o'clock.

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And Sunday, I think will be at 90 PM or something. So, and I'll announce that. So there's launch a chance to ask questions. Um.

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And I can maybe try a question or 2. now.

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Okay, and so so the office hours, so, Friday and Saturday are on a wiki I'll put the, um, Saturday 1 of, uh.

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Just a 2nd, here come on.

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Pull up the notes here.

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So.

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Okay.

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So, um, what we're talking about moving into basically finish up for. So now we're talking.

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After 5 and its pairs.

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Of random variables so your experiment might be to take RPI student and, uh.

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So, the random experiment.

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They pick an student and move on to measure.

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Wait wait. Okay.

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Called or something so it got 2 random variables.

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Capital letters random variables now um.

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For a single variables, you could talk about stuff like, expected value of height.

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Expected value of weight, standard, deviation of hide and, you know, you've got distribution function, density function, et cetera. Okay. Um.

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Now, for 2 variables.

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You got all the stuff for 1 variable, but then you've got.

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There are now joint statistics. Okay.

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Okay, and discreet and continuous and mixed and so on. So, um.

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Azure needs example than how you can wait, let's say here, let's say our experiment.

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2 fair so excited.

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Nice okay. And the 2 numbers are.

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And you say X and Y, okay. And each number is 1 6.

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Now, okay, so for 1, you know, for 1 variable.

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So, you can talk about, say probability of X so.

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So that's the probability I saw next. So if it's a fair side and die, so it wouldn't be say the probability is saying that.

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2 would be say 16, I could talk about why.

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Oh, okay. Saying a 3 on the 2nd dice would also be 16 because I said they're fair, but now we can talk about joint.

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Probabilities so joint probability. So I might have probability.

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Say some sort of X and Y, let's say so probability at the 1st die show the next and the 2nd day showed away. So the 1 experiment is it talks the 2 days and look at them.

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And so it would be say, you know, a particular 1 might be 136 or something. It's fair.

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Now, um, you could get some weird things and say, let me make it even. So, let me do a simple experiment here. Um.

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And scroll again, if I, if I scroll too fast, let me know, let me do a simpler experiments.

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Experiment.

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Throw 2 coins and, um.

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So the 2 outcomes are, uh, so, you know, eh, okay, let me say, and we measure, um.

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Why the X equals 0 for tail.

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1, for head 1st, the 1st going. Okay. And why? 2nd, um.

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And then we got our joint thing. Yeah, we could even do a table or something like this. Um.

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X and y1 1 0 and 1.

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You know, it could even look something.

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Let me do something a touch weird. Let's say.

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Let's say, um.

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I don't know.

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No, no, no, I mean, it back off. Sorry let me see, how do I want to do this? I'm making something that go along.

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1, 6.

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Okay, so this is the, this is like, showing a probability mass function or something.

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And we can look in the table so probability that the 1st coin get a tail and also the 2nd coin get a tail was 1 6.

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Probably the 1st coin was ahead and the 2nd point was a tale.

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Was 4th and so on. So so the Pro so this case it's a discreet experiment. Um, outputs are accountable.

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Finite or either accountable infinite and if it's fine, we can have a table here and shows the 2 variables together. What happens is 2 coin each header tails. I'm ignoring the possibility that the coin might land on edge.

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Ignoring the possibility if I tossed it, it falls into a hole in the floor or something and I can't measure it.

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Okay, so, for before joint outcomes here, um, okay now.

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And for something simple like this, you could just, like, have a table that shows it.

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Um, now interesting thing here, of course.

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Is there's a new concept called a correlation.

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And that will X and Y are dependent on each other, the dependent random variables.

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That, um, because.

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X and Y, are likely to be somewhat.

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Different from each other, if you know that the 1st coin was a tale.

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Then the 2nd coin is more likely than not to be a head.

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Because it's the 1st coin is a tale then you're in the 1st column. X equals 0 means the 1st coins of tail.

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And why it's more likely to be a head. So there's a relation between the 2 variables. And this chapter 5 is 2 random variables. So you do an experiment, you get 2 variables out.

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Yes, oh, yes. This means that in this case, the coin is not a fair coin. Yes.

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Thank you let me put that down. That's a good point. Um.

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They're not fair points. Yeah.

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They're both not fair. It's 2 points we are tossing, so yeah.

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What does the relation? Also? The previous thing I started off the class talking about the, your experiment was to pick a random student and measure the height and the wait for the same student. They're going to be related a student. That's taller.

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Will be more likely to be way more, so okay.

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Any case this is the 2 variable probability mass function here.

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Oops, come on, so you can look up, like, say, probability, mass function. P and then you say the 1st coin being a 0 and the 2nd coin being a 1 you look into the table.

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And that would be 4th, let's say.

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Okay, now.

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You can sum up rows and columns.

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To get the mass function for 1 for either variable separately.

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Some rows or columns.

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Probably mass function, right? They're variable separately.

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So you're right the thing again 0, 1 0, 1.

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64th 4th what? 6.

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Okay, so, you know, we could come over here.

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And we said stop and get 1 half on half and some down the bottom. We get 1, half half.

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So, actually you would have, um, just for X.

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X being 0 would be 1 half actually.

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So, as also, the probability of X being 1 is also the profit of why being 0.

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Y, v1, so you asked her the coin fair or unfair it's a complicated answer in this case, because you look at each coin separately.

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The probability of the 1st coin being heads or tails is 5,150.

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The probability of the 2nd coin being heads or tails is 50, 50.

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The way they're unfair is they're somehow tied to each other. I don't know how you'd implement that, but, um, if it's a software thing with a computer program, of course, you could do that. Um.

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So you go to casino and use.

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Slot machine or something it's all a computer program like.

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Wheel spinning and so on, they're just graphics it's all being done by a computer so.

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You know, they could program the thing to do something crazy, but any case.

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We have the 2 variable probability, mass function, and we summit. We can get these marginal totals and we can then get the 1 variable probabilities.

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From the too variable thing, but if you look at each separate 1 variable thing, it doesn't tell you anything about the relation and the interesting the new stuff in.

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Chapter 5 is that.

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Is ways to look at the relation between the 2 variables to somehow.

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Yes, sign numbers to it. Okay. So the new point in today's structure so far is the 2 variables and the discrete variables that say they're finite.

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Discrete variables you can do this table of the, the probabilities of all of the pairs to variable probability mass function.

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Okay, now, for 1 variable, you've got a kill the distribution function. Um.

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So, for 1 variable.

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I have a okay.

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Solution function, and so for the coin here.

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And so basically, so the probability, the random arrows less. So if for the 1st coin.

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It would have probability that assessment 0 is 1 half.

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Probably less than or equal to 1 equals 1. so the CDF would look something like this. Um.

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This would be 0, this would be 1. so this.

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Okay, now you have the same thing for 2 variables. Um.

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And so this is the way we so this is the probability and this would be here.

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So, what a few of the function would look like very discrete variable. It takes a jump each time.

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That function changes. So the 2 you have 2 variable CDF.

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So, and this would be defined as a probability that.

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And a variable exclusive X, and.

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Why let's say it and it's described, it's defined for discrete. It's defined for continuous. We're talking about the screen so far.

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So, for the, um, again, so, for this 2 variable coins that are tied together in some unknown way.

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And again, what they were were so, the density fund here, they probably called a mass function for the discreet case where we had 0 1. 0, 1.

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1, 6, 7th, 1, 6.

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So, if the kill, the function is, um.

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So the big 1, so the probability that both random variables arrested equal to 0, be 1 6.

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Well, you said another way, I could, I can draw the thing is like this.

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0 and 1 0 and 1, um.

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So this point here is probability 1, 6.

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It's point here, 4th, this point here on 3rd, at this point here 1, 6.

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So color the probability so the CDF and 00 is the probability or anywhere down in here.

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And That'll be that includes only the 1 point. So that's 1 6.

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The, they're on 1 that would be this here.

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That'll be 1+1 6+1.

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The cumulative density function for here.

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X and Y, say 10 will also be 1 half.

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And y1 and 1 would be 1 could be somewhere out here.

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So the way we extend it, the 1 variable killed, the function of 2 variables.

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Feel the function is we fix the point and it's the probability.

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That the random variable is anywhere below and to the left of that point, including that point.

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So, it, it gives the probability that the random variables somewhere in this quarter plane.

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So.

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I'll give you some more examples to think about that, but.

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So, of course, the is defined for any point in the plane. And so the way that it works out is if I sketch this thing again. Um.

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Like, color here oh, that's too light. Um.

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2nd, here darker.

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And again, at a point here, um.

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Point here to the value 1 6.

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He said this is the mass function. I'm drawing these points. Okay.

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So, the CDM, um.

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Again, so the way the CDF works is, if you're anywhere down in here, the is going to be 0 so.

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Less than strictly less than 0. I'm sorry. Um.

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Does a 0, um.

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And anywhere up here, let's say.

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Um.

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Well, actually affects lessons there or wireless than so it's going to be 0, let me write that down better. Um.

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Okay, um.

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0, if you're anywhere here, the probability.

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That X and Y, or, you know, loans at this point is 0, because it's not possible because X and Y are non negative. So.

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So, anywhere down in here, the is going to be 0, let's say.

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If you're up in, um.

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Let's say you're up somewhere like here.

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Going to be a 6 and so on. So we can divide the plane in the regions in in each region.

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Figure out what the CDF is um.

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Let me do that for you maybe. Oh, let me take this to the next stage here.

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So, here we got a 00 is a probability 1601 to the primary 4th.

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So, we could divide the whole plane enter.

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Are you good to Julie and.

189
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So, what are our probabilities if we're down here it's going to be.

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They're all so, what this means is f, X and Y equals 0.

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If 0 and wireless, it's also 0 here in here over here and 0 here.

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And we're up here is 1, cause it affects a lot greater than 1 the problem being that it's 1, it gets more interesting if we're in somewhere like here.

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So effects and wire in this region. What's the CDF.

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It's actually going to be 1, 6.

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And over here, it will be 1 and a half and over here will be a 1.

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I got it right so, basically, if, um, 0, less than X, less than 1.

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And there are lists Hawaii less than 1.

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For X and Y equals 1 6.

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And so on, so we can get for different regions.

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Um, 0, less and less than less an infinity materialist Y, less than 1.

201
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F X and Y equals 1 yeah.

202
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And so on, so it gets a little more complicated, but 2 dimensions are more complicated than 1 dimension. So.

203
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It's a 2 dimensional CDF.

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Now, the book use a little more complicated 1 where we've got to dice.

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Which are tied to each other in some unknown way.

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So each day separately has a 16th chance of having each 1 of the 6 possible faces is showing.

207
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Each time separately is fair, but somehow the chance of a pair.

208
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The 2 dice showing the same number is higher than it should be. So, the 2 dice together.

209
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They're tied together in some weird way.

210
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Oh, so let me write that down. Um.

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2 separately.

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They're fair, but your joint probabilities.

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Are not fair so.

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I just cut it down from dice. Yes.

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Yes, a little bit.

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That are already.

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Why, why is it?

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Yes, also the.

219
00:28:49.439 --> 00:28:55.709
Um, you know.

220
00:28:55.709 --> 00:29:00.479
Thank you, um.

221
00:29:01.558 --> 00:29:11.578
Yeah, you caught a typo and that's another email me. You could a point for that 1. that's.

222
00:29:11.578 --> 00:29:15.659
There you go, that was an non trivial. Thank you.

223
00:29:15.659 --> 00:29:28.048
Okay, so again, you catch me in a non trivial error might give you a point. So okay. You just got 1. okay.

224
00:29:28.463 --> 00:29:42.804
Yeah, in any case, let's go back to the 2 coins thing and again, so just to remind you 0 and 1 events here.

225
00:29:47.818 --> 00:29:51.388
10 and 106. 4th.

226
00:29:51.388 --> 00:29:57.659
6 or the 3rd. Okay. So the expected value of X.

227
00:29:57.659 --> 00:30:04.979
Equals 16 times 0+4+4th and so on.

228
00:30:04.979 --> 00:30:10.259
Accepted value why it was 4th cause it's the metric.

229
00:30:10.259 --> 00:30:13.378
Expected value of X squared.

230
00:30:13.378 --> 00:30:19.528
Equals on 6 times. 0+4th times 1 equals.

231
00:30:19.528 --> 00:30:25.229
4th, also the variance of X.

232
00:30:25.229 --> 00:30:29.368
Is expected value of X squared minus expected value back.

233
00:30:29.368 --> 00:30:35.219
Squared equals, uh, my 3-the 9th.

234
00:30:35.219 --> 00:30:39.868
Call 29 I did it right?

235
00:30:42.298 --> 00:30:48.838
So, okay, we can find variances for thing, goes for single things separately.

236
00:31:03.388 --> 00:31:13.979
Now, we can also, um.

237
00:31:13.979 --> 00:31:17.368
Fine things together and.

238
00:31:20.459 --> 00:31:26.338
Things like.

239
00:31:26.338 --> 00:31:30.148
Get the page number here, whichever.

240
00:31:39.929 --> 00:31:43.019
Trying to get the right page number. Um.

241
00:32:08.094 --> 00:32:11.483
This will be a new concept here is to call the CO variance.

242
00:32:12.358 --> 00:32:17.999
On page 258 here.

243
00:32:23.818 --> 00:32:28.618
And jumping around before, so the code variants um, well.

244
00:32:28.618 --> 00:32:32.189
Measure in a sense how they relate to each other.

245
00:32:32.189 --> 00:32:39.179
And the code variants of X and Y.

246
00:32:40.949 --> 00:32:45.179
And being defined as the, um.

247
00:32:48.269 --> 00:33:01.259
Excuse me? Okay.

248
00:33:01.259 --> 00:33:06.298
Variance of X and X is the variance. So.

249
00:33:13.469 --> 00:33:20.608
We've been talking too much today. Um.

250
00:33:20.608 --> 00:33:24.749
But the code variants will relate to how they, um.

251
00:33:26.159 --> 00:33:29.578
If they grow and shrink together.

252
00:33:29.578 --> 00:33:33.808
And then the CO variants will not be 0. so, um.

253
00:33:35.519 --> 00:33:41.308
Larger, and why go together.

254
00:33:46.078 --> 00:33:56.638
And the coherence of X and Y, greater than 0.

255
00:33:56.638 --> 00:34:02.308
Effects and why move in opposite directions.

256
00:34:05.098 --> 00:34:14.818
Directions, um, then the CO variants of X and Y, and less than 0.

257
00:34:14.818 --> 00:34:22.199
And if they have some complicated nonlinear relationship that.

258
00:34:23.608 --> 00:34:29.909
We can't say, but this, um, captures only linear relationships. Um.

259
00:34:29.909 --> 00:34:35.039
Better.

260
00:34:35.039 --> 00:34:38.969
This captures.

261
00:34:40.858 --> 00:34:50.489
Linear relationships, um.

262
00:34:51.568 --> 00:34:57.389
Let's say, um, say why it was X squared.

263
00:34:59.099 --> 00:35:02.789
That's a perfect relationship. We know what we have actually know why.

264
00:35:02.789 --> 00:35:06.148
You know, the covariant.

265
00:35:08.278 --> 00:35:15.989
0, perhaps, um, but there's a strong relationship. Um, okay.

266
00:35:22.829 --> 00:35:29.009
So, these things are going through their measure, detecting and measuring linear relationships.

267
00:35:29.009 --> 00:35:32.099
Not complicated relationships, um.

268
00:35:32.099 --> 00:35:41.458
So, give you another example, let's example, 2 um, let's say square it plus Y squared equals 1.

269
00:35:41.458 --> 00:35:54.268
This is what it's going to look like the code variants will be 0, but just a strong relationship. Okay. Okay. So, um, any case we define the covariant um.

270
00:35:57.148 --> 00:36:00.989
So, I can compute it for that, uh, to coin case.

271
00:36:07.858 --> 00:36:11.278
Uh, okay, so the 2, the 2 coin case.

272
00:36:11.278 --> 00:36:20.548
And again, the CO, and why it goes expected value of X, minus expected value of, um.

273
00:36:22.289 --> 00:36:33.268
I get it right here. Um, no, let me fix that up.

274
00:36:33.268 --> 00:36:38.668
And did I get it right up here?

275
00:36:38.668 --> 00:36:45.929
Actually, I had my parenthesis being wrong up here. Sorry I, I'm just going to take that out.

276
00:36:45.929 --> 00:36:50.789
Could I have branches? That's what I thought. Okay.

277
00:36:50.789 --> 00:36:54.989
Oh, rewrite it. Oh, you can get this so.

278
00:36:56.759 --> 00:37:00.208
So the CO variance is.

279
00:37:02.639 --> 00:37:06.539
Expected value X minus expected value X.

280
00:37:06.539 --> 00:37:10.139
Times nice the value of why.

281
00:37:10.139 --> 00:37:19.199
So our 2 coin case, um, 0 and 10 and 1164th 4th, 1, 6.

282
00:37:20.789 --> 00:37:29.248
Um, and so you just add the add the poor cases basically the equals.

283
00:37:30.329 --> 00:37:35.668
Um, well, the 1st thing is that we have.

284
00:37:40.648 --> 00:37:46.409
Yeah, the 2nd value acts as a half. Well, I mean, I'll rewrite the thing. It's the expected value.

285
00:37:46.409 --> 00:37:53.338
X -1, half wine -1 half I just plugged in and they expected values.

286
00:37:53.338 --> 00:37:58.168
Um, so we do the 4 things. Um, we have equals 1, 6 0.

287
00:37:58.168 --> 00:38:03.329
And if X and y0 0, that'd be 1, 6 times.

288
00:38:03.329 --> 00:38:08.969
-1 half -1, half +4th times.

289
00:38:08.969 --> 00:38:12.929
X is 1 and why is 0 So that will be.

290
00:38:12.929 --> 00:38:16.199
1, half -1 half.

291
00:38:16.199 --> 00:38:24.780
+4-1halftime is 1 and a half.

292
00:38:24.780 --> 00:38:29.639
+1 6 times, um, on half.

293
00:38:29.639 --> 00:38:34.739
1, half and that will be.

294
00:38:40.199 --> 00:38:46.289
Quarter time the 624.

295
00:38:47.639 --> 00:38:53.460
We have - minus, um, 1, quarter times a 3rd.

296
00:38:54.840 --> 00:38:58.889
112-112.

297
00:38:58.889 --> 00:39:03.630
On 24th, um.

298
00:39:03.630 --> 00:39:08.969
On 12-212-112, it's negative.

299
00:39:08.969 --> 00:39:13.199
So, they have this opposite relationship to each other.

300
00:39:15.599 --> 00:39:20.639
Okay, any case we can compute things like that.

301
00:39:20.639 --> 00:39:26.340
So, yes.

302
00:39:28.110 --> 00:39:31.469
All the various.

303
00:39:33.900 --> 00:39:37.590
So this 1.

304
00:39:40.320 --> 00:39:47.489
Uh, talk to sign up for the negative.

305
00:39:48.594 --> 00:40:02.155
Okay, there's 4 terms. The 1st, 1 is 2 negatives. The 2nd 1 is 1 the 3rd 1 is 1 and the 4th 1 has no negatives and I'm adding the 4th. Well, we've got 4 cases here with the 2 coins, tails, tails, tails, head, head, tails and had had.

306
00:40:02.639 --> 00:40:11.010
Tails tails has probability 16th and in that case, Texas, well, the expected value back to the half.

307
00:40:11.010 --> 00:40:15.840
Okay, so if we look at this thing down, um.

308
00:40:16.860 --> 00:40:23.940
Yeah oh, you're right. Okay. Thank you. I'll just.

309
00:40:25.019 --> 00:40:31.139
Uh, oh, there is.

310
00:40:31.139 --> 00:40:35.849
Sorry about that. Yeah, that's a minor thing. It's not worth the.

311
00:40:43.050 --> 00:40:49.409
So, what we'll do later is we'll take the code variants and expand it to something called a correlation coefficient.

312
00:40:49.409 --> 00:40:54.929
Which will be a dimension less number, which will try to capture the relations. So.

313
00:41:15.985 --> 00:41:23.034
No, there's also a chance to think back to dependent versus independent variables. We can take the 2 of them and.

314
00:41:23.280 --> 00:41:27.719
And the shoulder dependent on each other. So if we remember that, um.

315
00:41:29.639 --> 00:41:37.860
Wire independent, if it only if.

316
00:41:37.860 --> 00:41:43.769
Probability of some particular value occurring together like.

317
00:41:44.880 --> 00:41:49.619
Vertical something, and why being some value is the.

318
00:41:49.619 --> 00:41:57.510
It's the probability Time's a probability. Why why let's say.

319
00:41:57.510 --> 00:42:02.579
That's the definition of independence now, in this case, the 2 coin case.

320
00:42:06.420 --> 00:42:10.590
So, the probability of X equals 0 and why those arrows 1 6.

321
00:42:10.590 --> 00:42:15.960
For the probability of X equals 0 ignoring why is 1 and a half.

322
00:42:15.960 --> 00:42:20.190
Which is also the probability of why Voicera ignoring X.

323
00:42:20.190 --> 00:42:32.880
But 16 is not equal to 1, half times 1, half so, external or independent so, or would say not so we can bring back stuff from the previous chapter here.

324
00:42:37.469 --> 00:42:42.510
And, um, okay, so.

325
00:42:51.329 --> 00:42:56.940
So, we can also go back and look at bring some stuff.

326
00:42:56.940 --> 00:43:05.130
From chapter 4 forward into chapter 5, look calculate they say the standard deviations and so on. So.

327
00:43:06.869 --> 00:43:11.369
Oh, so we can say, well, new.

328
00:43:11.369 --> 00:43:18.960
Value of X what happens on why it was 1 half? Um.

329
00:43:18.960 --> 00:43:25.349
And again, the Sigma Sigma of X, uh, was again to square root of the variance.

330
00:43:28.139 --> 00:43:37.500
And the variance will re, computed at the variance as expected value of X squared device.

331
00:43:38.760 --> 00:43:43.289
Squared and again, um, the variance.

332
00:43:43.289 --> 00:43:50.309
Was expected value that squared was if I were.

333
00:43:50.309 --> 00:43:56.280
Would be 1, half -1 quarter.

334
00:43:58.619 --> 00:44:07.650
So, the, uh, sandbox was 1 here also. Okay. Oh, that's what signal Y okay.

335
00:44:07.650 --> 00:44:12.630
Okay, now we can define a new term. So here's a new.

336
00:44:12.630 --> 00:44:17.909
A new term. Oh, the correlation coefficient um.

337
00:44:26.489 --> 00:44:35.789
And that is defined as, uh, the CO variants of X and Y.

338
00:44:35.789 --> 00:44:39.090
Divided by the 2 standard deviation.

339
00:44:40.469 --> 00:44:48.269
I put the thing and highlighted perhaps. Okay.

340
00:44:48.269 --> 00:44:54.809
Now, the nice thing with the correlation coefficient is it's dimension list.

341
00:44:54.809 --> 00:45:00.900
So so it goes from -1 let's go to a row.

342
00:45:00.900 --> 00:45:04.289
The wireless at 1, 2 1.

343
00:45:04.289 --> 00:45:11.130
So, I mean, let's suppose you're measuring, you know, heights of 2 stone types and waits for something. So.

344
00:45:11.130 --> 00:45:16.199
Are you doing metrics to the height is meters in the way it is kilograms.

345
00:45:16.199 --> 00:45:25.230
Uh, and, um, you'll get a certain Co variance and then if you switch to English imperial units, height being and feet and weight being in pounds.

346
00:45:25.230 --> 00:45:30.840
The comparisons will be different because the variance depends on the units just like, um.

347
00:45:30.840 --> 00:45:34.829
You know, I mean, you know, the main height is say 5 feet.

348
00:45:34.829 --> 00:45:43.230
It's got the unit, um, or 1.9 meters now with the correlation coefficient we canceled out the units.

349
00:45:43.230 --> 00:45:48.090
So, the correlation coefficient does not depend on.

350
00:45:48.090 --> 00:45:54.269
On the unit, it goes from -1 to 1 absolute value with correlation coefficient less than 1.

351
00:45:55.530 --> 00:46:01.260
And what it means is that if the correlation coefficient is 1.

352
00:46:01.260 --> 00:46:10.980
Then there's a perfect positive relationship between X and Y, if it's -1 is a perfect negative relationship.

353
00:46:10.980 --> 00:46:15.570
If it's 0, there's no linear relationship either way.

354
00:46:15.570 --> 00:46:25.949
There could be a complicated nonlinear relationship like experts with white square equals one's a beautiful example. Correlation. Coefficient would be 0 in that case.

355
00:46:25.949 --> 00:46:34.530
So all right, some of this down, um.

356
00:46:35.789 --> 00:46:41.070
So, row X, Y, equals 1. that's a role. If you can't read my handwriting.

357
00:46:43.139 --> 00:46:47.969
The perfect positive relationship.

358
00:46:52.199 --> 00:46:56.730
The relation probably put -1.

359
00:46:56.730 --> 00:47:05.730
I got it. Okay. Well, example, perfect positive might be say why it was 5 X. +3.

360
00:47:05.730 --> 00:47:15.510
It'd be a perfect positive 1 say, probably post 1. why it was -2 X. +1. That'd be really plus -1. so.

361
00:47:15.510 --> 00:47:19.320
I don't know, it goes 0, no linear relationship.

362
00:47:23.429 --> 00:47:28.980
Okay, so correlation coefficient sometimes get into the popular press.

363
00:47:28.980 --> 00:47:34.829
As there's a correlation between something and something else, and this is what they're talking about. So.

364
00:47:34.829 --> 00:47:45.750
Um, role has to be fairly large for the relation actually, to be interesting. So.

365
00:47:52.650 --> 00:48:00.659
Arch, um, okay, Gordon interesting relationship.

366
00:48:04.829 --> 00:48:09.420
That'd be interesting. Um, so.

367
00:48:10.650 --> 00:48:17.309
I don't know, basically, it has to be much greater than a half.

368
00:48:17.309 --> 00:48:25.650
So somebody set a correlation coefficient of a quarter or something and that's not interesting. It's probably due to chance something. So.

369
00:48:29.699 --> 00:48:35.519
Okay, in any case. So we've seen the definition of the code variants between 2 variables.

370
00:48:35.519 --> 00:48:42.210
But 2, variables of being measured on the same in China, we say in the correlation coefficient, which was the.

371
00:48:42.210 --> 00:48:46.320
Coverage where you canceled out the dimensions.

372
00:48:46.320 --> 00:48:59.099
Questions about that and you could prove it's absolutely last article to 1, but I'll do that today. So.

373
00:49:01.860 --> 00:49:05.789
Okay.

374
00:49:08.550 --> 00:49:12.389
Example of where you're trying to do this, um.

375
00:49:15.420 --> 00:49:18.750
Is that you actually might have more than 2 variables.

376
00:49:18.750 --> 00:49:21.900
And you're doing correlation coefficients.

377
00:49:21.900 --> 00:49:26.309
Well, what happens sometimes that people are trying to do some prediction.

378
00:49:26.309 --> 00:49:30.059
I tried to predict horse racing, let's say.

379
00:49:30.059 --> 00:49:37.050
They've got lots of variables. You've got the horses average. Hey, you've got how many dollars at 1.

380
00:49:37.050 --> 00:49:43.110
Previous races, you know, where it came and you've got and, you know.

381
00:49:43.110 --> 00:49:48.300
Did it went a race to find any correlation coefficient say.

382
00:49:48.300 --> 00:50:00.480
Between the horse, the horse, how many races of horse? 1, and well, with an X Ray, you got a correlation coefficient and you may look at how many dollars the horse 1.

383
00:50:00.480 --> 00:50:04.289
Versus when the next race, you get a correlation coefficient.

384
00:50:04.289 --> 00:50:13.380
And, of course, they're not the same because the horse might have won a lot of cheap races and no expensive races, expensive races of the biggest higher quality horses. So the persons would be higher.

385
00:50:13.380 --> 00:50:18.869
Okay, so maybe you're trying to find a betting scheme so you've got a Saratoga.

386
00:50:18.869 --> 00:50:21.989
In August and make some money.

387
00:50:21.989 --> 00:50:27.659
So, you'd be, you know, you'd be buying statistics and you'd be doing correlations like that.

388
00:50:27.659 --> 00:50:31.409
And what you might have is actually try to find.

389
00:50:31.409 --> 00:50:35.579
The input variables would be called on independent variables. So.

390
00:50:35.579 --> 00:50:39.239
How many races? The horse is 1. how many dollars? It's 1.

391
00:50:39.239 --> 00:50:50.130
It's ranking and that's sort of thing. Um, and then the outputs to try to predict would be the dependent variable. It's dependent on the independent variable. So we've been doing correlation coefficients.

392
00:50:50.130 --> 00:50:57.119
And we trying to find independent variables that had the larger correlation coefficients with the dependent variable. So.

393
00:50:57.119 --> 00:51:01.829
Something like the horse thing, maybe.

394
00:51:01.829 --> 00:51:05.130
Who knows the number of dollars it's 1 might be.

395
00:51:05.130 --> 00:51:10.559
A better predictor of today's race and how many races it's 1 in the past because.

396
00:51:10.559 --> 00:51:20.699
Dollars it's 1, man, it's winning against good horses races its 1, or may have been raised against bad forces. But in any cases he'd be doing these statistical measurements.

397
00:51:20.699 --> 00:51:28.349
Same thing for the stock market. Um, so this is a place where the correlation coefficients come in and in fact.

398
00:51:28.349 --> 00:51:35.880
It might even be doing something called multiple stepwise, linear regression where you've got a lot of possible independent variables.

399
00:51:35.880 --> 00:51:43.320
They're trying to find the ones that are most important so it'd be looking at lots of correlation coefficient. So that'd be an application of this. So.

400
00:51:43.320 --> 00:51:52.230
Actually, when I was in grad school, 1 year, 1 of my roommates, and I, we started playing and trying to predict things with the horse races.

401
00:51:52.230 --> 00:51:57.719
And I like the mathematics, but my roommate actually, like, going looking at the horses. So.

402
00:51:57.719 --> 00:52:01.469
Is a is a fun to learn some applied statistics. So.

403
00:52:01.469 --> 00:52:14.670
What we did is we advertise and we bought some racing forms. That's the daily newspaper that gives statistics for racing there. This was before you could buy it on computer. So we bought like, a 6 foot high stock, erasing forums and.

404
00:52:14.670 --> 00:52:18.780
We typed in 2000 lines of data. Um.

405
00:52:18.780 --> 00:52:23.699
There are punch cards and I started doing these sorts of things, correlations and selling for fun.

406
00:52:24.869 --> 00:52:32.579
There was an assistant dean of engineering called some use that I used to go to Saratoga on the week in August and I think he.

407
00:52:32.579 --> 00:52:36.989
Said he made some money, and it was also chairman of the computer science department. Um.

408
00:52:36.989 --> 00:52:44.369
At 1 point, also go off to Saratoga and said he could make money doing this sort of stuff. So.

409
00:52:45.690 --> 00:52:49.829
You know, so who says, you know, the courses are not relevant.

410
00:52:50.664 --> 00:53:02.215
Okay, that was the stock market. Some people say that they, you know, they make it work.

411
00:53:02.994 --> 00:53:06.985
Um, you suspect that the people that really make it work are perhaps not talking about it. So.

412
00:53:08.610 --> 00:53:20.849
But MIT students did, um, do statistics to crack the Massachusetts state lottery some years ago legally so they were doing correlations and so on. Okay.

413
00:53:25.289 --> 00:53:28.650
The next thing we might say, have continuous variables.

414
00:53:32.460 --> 00:53:38.250
So, let's say X is uniform.

415
00:53:38.250 --> 00:53:43.110
So, in 1, so here, what this means is the density function.

416
00:53:44.880 --> 00:53:53.489
Equals 0, X less than 010 X less vehicle 210 X trader than 1.

417
00:53:53.489 --> 00:53:57.599
So, the density the cables are saying.

418
00:53:57.599 --> 00:54:04.139
It's going to look like that so it's 0 X less than 0 X less than.

419
00:54:05.280 --> 00:54:10.829
0 X, less than equal 11 if X greater than 1. so it's like that.

420
00:54:11.880 --> 00:54:16.860
And let's say, why is the same? So, let's just say they're defending and we might say.

421
00:54:16.860 --> 00:54:23.190
For why is the same say? It's, it's like FMX and so on. Okay.

422
00:54:28.920 --> 00:54:33.719
And let's say X and wire independent. Okay.

423
00:54:34.800 --> 00:54:39.900
Or today. Okay, so now you can get a joint.

424
00:54:41.550 --> 00:54:47.880
A joint density function or PDF probably that's a function. Have.

425
00:54:49.739 --> 00:54:54.300
So, the subscript is the random variables, the argument or the particular values.

426
00:54:54.300 --> 00:54:58.349
And I'll put that on the next page, so.

427
00:54:58.349 --> 00:55:03.599
Copying it so.

428
00:55:03.599 --> 00:55:07.800
So, the next, so, in this case, half of X and Y.

429
00:55:07.800 --> 00:55:14.730
Equals 1, if.

430
00:55:17.010 --> 00:55:23.190
And there's 0 else, otherwise.

431
00:55:23.190 --> 00:55:27.269
So, if we potted it, it would be, uh.

432
00:55:30.329 --> 00:55:34.829
Everywhere in here, it's 1.

433
00:55:34.829 --> 00:55:44.670
And everywhere outside, it's 0, that's the density function. Now what this means, um, the definition.

434
00:55:44.670 --> 00:55:48.329
So, the probability that random variable is.

435
00:55:48.329 --> 00:55:52.199
Between some X and some X .

436
00:55:52.199 --> 00:56:01.590
And also why is, um, the density function.

437
00:56:01.590 --> 00:56:07.710
Types D, Y, so for very small.

438
00:56:07.710 --> 00:56:19.829
For small DX and D. Y. okay. Yeah, that's definition of density function. In fact, 1 variable 2 guards. The same thing.

439
00:56:19.829 --> 00:56:23.489
So, it's 1 in that rectangle Carol. That's right.

440
00:56:28.440 --> 00:56:34.860
All of these things assume of the functions are smooth and don't do crazy, pathological things.

441
00:56:34.860 --> 00:56:42.630
That are beyond this course. So okay, so the, um.

442
00:56:42.630 --> 00:56:53.880
So, the chemical function again, so that's the probability that the ran and.

443
00:56:53.880 --> 00:56:58.739
Or intersect, I mean, it's the same thing. Why why.

444
00:57:00.179 --> 00:57:03.840
And what that's going to be is, um.

445
00:57:06.690 --> 00:57:16.409
Complicated mess. Okay. Um.

446
00:57:18.840 --> 00:57:28.829
Well, if X or Y are less than 0 than the answers 0, I haven't got a different color here. So here, it's 00000.

447
00:57:28.829 --> 00:57:33.179
If we're in here, 6 times, why.

448
00:57:33.179 --> 00:57:37.349
If were to the right where X is critical to 1.

449
00:57:37.349 --> 00:57:44.550
And then it just is why it just is excellent just as 1 up here.

450
00:57:45.809 --> 00:57:51.719
I got it, right? Yeah.

451
00:57:51.719 --> 00:58:01.949
Okay um, so I'll give you an example.

452
00:58:04.110 --> 00:58:09.599
Say, let's suppose access to and why is on 3rd.

453
00:58:09.599 --> 00:58:14.159
That's the probability that X is actually equal to 2.

454
00:58:14.159 --> 00:58:22.050
And while I listen for the 4th, well, X is always because X is a.

455
00:58:23.460 --> 00:58:28.889
It's in the range 0 to 1. that's probably wireless. Go to 4th.

456
00:58:28.889 --> 00:58:32.849
That'll be 4th, so.

457
00:58:34.110 --> 00:58:37.530
Because as the aside, let me just put that in actually.

458
00:58:37.530 --> 00:58:40.739
Probability of access vehicle to 2 equals 1.

459
00:58:40.739 --> 00:58:49.260
So, okay, so these things get to be complicated.

460
00:58:50.489 --> 00:58:59.250
Um, things so, but any case so, the cube distribution function for the.

461
00:58:59.250 --> 00:59:03.000
2 variable case, so it's uniform.

462
00:59:03.000 --> 00:59:12.090
Problem is for 1 variable. You may have 3 cases X less than 0 x0 to 1 x greater than 1.

463
00:59:12.090 --> 00:59:16.320
2 variables to get 9 cases so it gets to be Messier, right?

464
00:59:16.320 --> 00:59:20.400
That's like, oh.

465
00:59:20.400 --> 00:59:24.030
Well, that's how.

466
00:59:26.909 --> 00:59:31.980
Um.

467
00:59:38.250 --> 00:59:51.449
Now, how would you use this? Um, so you can find the probability of the random variables being in any rectangle by adding and subtracting the kill to function.

468
00:59:51.449 --> 00:59:56.309
So, um, let's just talk about for 1 variable.

469
00:59:58.530 --> 01:00:03.420
Probability that say excess from some low variable.

470
01:00:03.420 --> 01:00:10.230
To some high variable say, load a high that's, um, high minus the low.

471
01:00:11.400 --> 01:00:16.320
Okay, so, for this particular thing here, the probability that say.

472
01:00:16.320 --> 01:00:20.309
4th last it was X less than or equal to, um.

473
01:00:20.309 --> 01:00:28.170
3 quarters would be the able to function on 3 quarters minus the Cuban, the function on 4th.

474
01:00:28.170 --> 01:00:31.590
+3 quarters -4th.

475
01:00:31.590 --> 01:00:38.730
Well, uh, 912-412, 812, uh, 512.

476
01:00:38.730 --> 01:00:41.940
Okay, that's for 1 variable.

477
01:00:41.940 --> 01:00:45.630
Now, for 2 variables, you can do a similar thing. Um.

478
01:00:46.980 --> 01:00:54.510
So, the probability that, um, well, let me low and try and get to our debrief. So, let's just say.

479
01:00:56.579 --> 01:01:06.960
0, less than X, less than X1 and say, why 0 less than equal to Y, big Y that's why 1.

480
01:01:06.960 --> 01:01:12.659
That's then, so what we have here, if I f*** things out.

481
01:01:12.659 --> 01:01:23.969
So, I want to find the probability. This is a point x0 y0. This is the point. X1. y1. Okay and this is excellent.

482
01:01:23.969 --> 01:01:28.440
Why 0, and this is line 1.

483
01:01:29.670 --> 01:01:33.840
So, I want the probability. So what I do is that.

484
01:01:33.840 --> 01:01:37.650
I take the probability for this thing here.

485
01:01:41.519 --> 01:01:44.730
Plus the probability.

486
01:01:44.730 --> 01:01:52.769
Or this thing here, minus the probability for this thing here.

487
01:01:55.530 --> 01:02:00.449
Minus the probability of running out of colors for this thing here.

488
01:02:02.519 --> 01:02:07.710
And if you check it all at the end of all of this.

489
01:02:09.809 --> 01:02:13.949
What I've done is I have I have.

490
01:02:13.949 --> 01:02:18.179
And go to the square here once I've included this thing here.

491
01:02:18.179 --> 01:02:23.280
0, cause 1-1 equals 0 and put in this region 0 times.

492
01:02:23.280 --> 01:02:28.559
And this region here 0 times, cause I added it twice and I subtracted it twice.

493
01:02:30.420 --> 01:02:37.079
So so I can find the probability that they ran the variable.

494
01:02:37.079 --> 01:02:44.039
Hair is in that rectangle by adding and subtracting the cumulative functions for the 4 corners.

495
01:02:44.039 --> 01:02:53.519
So, um, no more coughing here so write that down. So, um.

496
01:02:53.519 --> 01:02:56.579
So, the probability again that the DX.

497
01:02:59.190 --> 01:03:10.710
And why is there why less than strictly? It was continuous it doesn't matter. Let's be strict. So that's going to be the field of function on X1 y1.

498
01:03:10.710 --> 01:03:14.789
Plus the came with a function on y0.

499
01:03:14.789 --> 01:03:18.840
Minus on y1.

500
01:03:18.840 --> 01:03:22.739
S1 0, so.

501
01:03:24.630 --> 01:03:28.650
And we could do this for a more complicated rectangular shapes.

502
01:03:28.650 --> 01:03:33.239
Now, what, if you want to do it for a shape that's not a box. That's not a rectangle.

503
01:03:33.239 --> 01:03:36.900
You know, you do some sort of enter girls. Okay. So.

504
01:03:38.940 --> 01:03:46.440
Okay, and again reminder why we use CDs. Let me write this down. Um.

505
01:03:54.599 --> 01:04:00.869
Well, I use the cumulative yeah.

506
01:04:00.869 --> 01:04:06.000
It works for both discreet.

507
01:04:06.000 --> 01:04:14.639
And continuous distributions, uh, continuous.

508
01:04:16.440 --> 01:04:21.960
Distributions and mix also.

509
01:04:23.039 --> 01:04:30.750
And mixed distributions also supposed to be.

510
01:04:34.320 --> 01:04:40.050
Okay, so, um, that's why it's nice.

511
01:04:41.610 --> 01:04:46.650
Again, what would be an example of a mixed distribution? Um.

512
01:04:46.650 --> 01:04:51.449
You arrive at the airport, you want to pick up a car to drive you um.

513
01:04:52.650 --> 01:04:55.739
Here back to our API.

514
01:04:55.739 --> 01:05:04.170
Well, the reasonable chance is a car already. There is an Uber is a taxi or an Uber already circling there. So you're waiting time is Daryl.

515
01:05:04.170 --> 01:05:11.610
If you're waiting time is not 0, that it's going to be some exponential distribution with the mean of 5 minutes or whatever.

516
01:05:11.610 --> 01:05:17.190
So that's a mixed distribution. There's a probability of a specific point. Timing was 0.

517
01:05:17.190 --> 01:05:22.260
And then the probability of it being smeared out over a range. So.

518
01:05:27.119 --> 01:05:31.949
Let me just remind you of what the mixed distribution is useful for. So I can.

519
01:05:31.949 --> 01:05:40.889
So next example.

520
01:05:43.110 --> 01:05:46.409
Get a car at the airport.

521
01:05:51.900 --> 01:05:55.920
There it's already there.

522
01:06:00.090 --> 01:06:07.710
And then if not, then I probably have some time key is, um.

523
01:06:11.070 --> 01:06:18.840
The minus T or some scale factor and that's the scale factor there to make it add or add up to 1.

524
01:06:18.840 --> 01:06:23.909
Uh, well, that would be a density actually, so something like that.

525
01:06:25.800 --> 01:06:29.340
Yeah, so.

526
01:06:31.920 --> 01:06:35.429
Okay, this would be for team greater than 0.

527
01:06:35.429 --> 01:06:41.429
See, exactly. Equal to 0 all that. Okay. That's a that's a natural example of mixed function in here.

528
01:06:43.079 --> 01:06:49.949
Okay um, so that's a nice introduction to chapter.

529
01:06:49.949 --> 01:06:57.360
5, what's new here is that we have 2 variables so we do 1 random experiment and we get.

530
01:06:57.360 --> 01:07:02.400
2 variables to, you know, call them outcomes, whatever and.

531
01:07:02.400 --> 01:07:05.610
The thing is that the coming.

532
01:07:05.610 --> 01:07:10.349
Outcomes from the same, it's fair, but they may be tied to each other in some way.

533
01:07:11.400 --> 01:07:20.849
So, we obviously we could do calculations with each variable separately. We could compute its mean computer variance standard deviation.

534
01:07:20.849 --> 01:07:27.030
It's discrete we could look at its probability mass function. If it's continuous, it's density function.

535
01:07:27.030 --> 01:07:32.039
In both cases, we can look at the cumulative distribution function.

536
01:07:32.039 --> 01:07:40.289
As for 1, but for the 2 variables, since they may be tied to each other. In some sense we want to have.

537
01:07:40.974 --> 01:07:55.914
Things that work with both variables and if it's discreet, we could just do a little table there. Like, I pick your the example of 2 coins, which are, in some way, tied to each coin. Separately was fair, 50, 50 heads tails.

538
01:07:56.159 --> 01:08:00.989
But somehow they came up with opposite.

539
01:08:00.989 --> 01:08:08.070
Thing so the 1st coin showed ahead more likely than not. The 2nd coin would show a tale text you to 1.

540
01:08:08.070 --> 01:08:15.869
So you could, so if they're discreet things or say, find that number possibility, you could just do a table showing that.

541
01:08:15.869 --> 01:08:23.760
Um, if it's continuous, then you have some sort of equation for the dense can use density function.

542
01:08:23.760 --> 01:08:33.029
In any case, you could do a marginal sum and get the careful to function. In both cases. I showed you for the 2 coins.

543
01:08:33.029 --> 01:08:38.819
For these 2 variable discreet things. The kim's, a function is a mess.

544
01:08:38.819 --> 01:08:44.100
Yeah, it's like, um, be glad I used to coins and not 2 days.

545
01:08:44.100 --> 01:08:51.750
Like, the book use, but now you can start relating the 2 of them with something called the covariant.

546
01:08:51.750 --> 01:09:02.369
And I gave the formed expected value X, minus secondary times Y, minus expected value and his ways to rework that equation. We'll see them next Thursday.

547
01:09:02.369 --> 01:09:07.979
So the CO variants captures the linear relation between the 2 variables.

548
01:09:07.979 --> 01:09:19.979
It depends on the units, um, if you're looking at student Heights versus weights, if they're pounds and feet the number for the will be different than if they're kilos and meters.

549
01:09:19.979 --> 01:09:26.159
So, you can make a dimension less by dividing through by the standard deviations of the 2 variables.

550
01:09:26.159 --> 01:09:30.869
And that gives us a correlation coefficient. It goes from -1 to 1.

551
01:09:30.869 --> 01:09:40.380
If it's 1 is a perfect linear relation between the 2 variables. Like X equals 5. if wife was 5 X, +3, it's per false linear relation.

552
01:09:40.380 --> 01:09:44.699
If, um, it's -1 is a perfect negative relation.

553
01:09:44.699 --> 01:09:52.170
Usually scatter around is an imperfect relations, so there between 0 and 1.

554
01:09:52.170 --> 01:09:58.199
If there's a nonlinear relation, not beautiful examples export. Plus Y, squared equals 1.

555
01:09:58.199 --> 01:10:01.680
If, you know, X, you only got 2 values for why.

556
01:10:01.680 --> 01:10:07.710
So, they're really strongly related, but they're not linearly related. So the correlation coefficient.

557
01:10:07.710 --> 01:10:14.640
Will be 0, if X squared plus Y squared equals 1. so that's something to watch out for complicated relations. Don't get captured by this.

558
01:10:14.640 --> 01:10:21.750
Another thing is that the correlation coefficient has to be quite large. Like, I would say, greater than.

559
01:10:21.750 --> 01:10:24.810
50% at least or maybe even much greater.

560
01:10:24.810 --> 01:10:39.000
Or, to be interesting. So if somebody says they found a relation between 2 things, and it's correlate, you know, a correlation coefficient is point 1 or something, then you laugh because it's probably totally due to chance.

561
01:10:39.000 --> 01:10:42.060
Later in the course, can quantify that statement.

562
01:10:43.560 --> 01:10:49.949
And so what we'll get on more in the book chapter 5, we're working through these things in more detail.

563
01:10:49.949 --> 01:10:54.390
Show what's happening with the calcium distribution of course, cause.

564
01:10:54.390 --> 01:11:07.140
Curse so often, so that is a reasonable point to stop. If there's questions I'll think of answers. I hope.

565
01:11:07.140 --> 01:11:12.960
Have a good weekend. Um, we have, um, like this office hours tomorrow.

566
01:11:12.960 --> 01:11:16.109
Friday this office hours, Saturday.

567
01:11:16.109 --> 01:11:24.300
Virtual on Webex there's an office hours Sunday. I'll post it Friday and Saturday. 30. P. M. Sunday. 9. 0. P. M.

568
01:11:24.300 --> 01:11:30.119
And so you can ask lots of questions on Monday. We have a test.

569
01:11:30.119 --> 01:11:39.659
Here so have fun. It looks like, I guess it might be snowing tomorrow. So still being here.

570
01:11:39.659 --> 01:11:44.430
But get some exercise, go skiing if you don't know skiing, learn it.

571
01:11:44.430 --> 01:11:49.380
Good, you can study more effectively to get some physical exercise. Occasionally.

572
01:11:50.850 --> 01:11:55.050
So, I try to go skiing, whatever the snow out.

573
01:12:00.840 --> 01:12:07.890
Chris.

574
01:12:07.890 --> 01:12:14.279
Okay, cool.

575
01:12:14.279 --> 01:12:18.090
Sure, now I just curious, uh.

576
01:12:18.090 --> 01:12:21.899
I just wanted to see how people are in the court or meeting. So 1.

577
01:12:21.899 --> 01:12:25.380
Um, okay.