WEBVTT

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No good afternoon probability people. So this is the final class.

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26, I, thank you.

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You're ahead of me can see me and you can hear me. Can you also see my screen showing the blog for today?

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Yes, great, thank you. So, if you're curious what I finally did, I rebooted the computer. I applied updates I tried wasn't without the VPN. What I'm doing now is.

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It's different is that I'm using, I switched to Chrome from Firefox for the browser so Chrome used to work. It does not work.

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Now, all I can think that might have changed is there was some update, just continuous updates and something clashed. It might not even be Chrome.

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Who knows, I might be something in a bank.

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So this will be a short class.

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Just hit hit some random points for some. Some of you are interested in your grade and so there's no homework assigned today. The homework that was due today will be the last 1.

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Gray will be computed. I'll drop the lowest homework. Each homework will be normalized to be the same total weight as every other homework. The number of points the homework is out of is irrelevant. I mean, computers are good at doing arithmetic.

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I dropped the lowest homework.

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Exams of the 33 exams will be given the same weight and will drop the lowest exam.

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And then going to the cap using the waiting that I have in the syllabus and people have gotten to what all points get added in.

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So, what I'm going to do in a day or 2, once you've handed in the homework that's due today. Well, I guess it was due already. But and what I'll try to do tomorrow is I will calculate a guaranteed minimum, let her grade for you.

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And this will be the grade that you will get if you do not write the 3rd exam, the final exam.

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So, and then if you want to write it.

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Then good Elliot can do with race here grade at some point.

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In a few days,

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I might post a question somewhere seeing how many people want to write the final because it is a possibility that no 1 at all will want to write the final in which case.

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You know, why habit what's spend the time and we'll spend the time to set it.

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And also, if you're in China or East Asia, and you want to write the exam, 12 hours later, then drop me a note. This is.

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Only people in each stage are trying to get this opportunity so they don't.

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App to similar parts of the world.

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That part of had article would work also. Okay. Grading great either grading questions.

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Now, I have a chat window open, so I can see.

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What's happening on Thursday's video? Something didn't work is probably my fault.

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So, however, my good notes thing that I was writing by hand, I think, did work, I'll try to upload it as a PDF file.

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So, today, I'm nothing new to some review questions and I'll show you stuff, scan through stuff in the book and tidy up some loose ends and so on.

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You have seen some of these questions before, but who knows you might see them again. It tell me that environmental is is good.

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So, we might rerun something.

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If I go through number 1, say, what's the center girl here?

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Well, you might notice as this is the, um.

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Actually would apply to the calcium in the normal distribution, except for the and.

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The density function, you divide by 1 over Square, root too high. So.

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So, integral is going to be square root of 2 pie there. D.

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If you forgot this and you had access to Matt to Mathematica.

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Then integrated and Mathematica, or do a correlation coefficient the value ranges the minus 1 to 1.

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So.

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If we have an integrated circuit and.

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Some independent events cosmic re, stuffs, particles people sneezing. I don't know.

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Are causing defects.

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The most reasonable distribution is a croissant distribution.

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Um, and.

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You might say binomial would work with then being very large and K being very small.

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But it's it would work. You might have difficulty evaluating it. You'd have to use special techniques because offends very large and pictorials even larger. Um.

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But it wouldn't be reasonable.

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Normal would work if the mean is.

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Somewhat greater than 0, croissant.

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I mean, draw a few of them take advantage of the fact that, hey, we can actually.

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Share a screen, so that'd be a screen. Let's see if it works now.

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It worked cool.

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Always amazed when technology works.

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Such a pleasant surprise. Oh, so, if we're talking about to remind you what it looks like, um.

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If they meet the, if health is very small, and it looks something like that. Um.

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Alpha was a little bit bigger. It might look something like that.

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If always even bigger.

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Look something like that and so on so a normal distribution.

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Would be okay for the blue 1 with alpha being somewhat bigger and not so good for the red 1 and totally inappropriate for the black 1.

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Okay.

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And again, it's when you have a large number of possible events, but the probability of any 1 event is very small.

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So isn't appropriate. Let me write that down. I mean, we've done this before and before before, but it's always good to nail things home. Occasionally a 2nd, here.

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Until the next event hits your web and that would be exponential. And the point of an exponential.

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I just make sure I've got the chat window open again. The point about exponential is it's memory. Let me just put a note about this here.

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Silence.

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The time to the expective next event does not depend on how long you've been waiting.

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Okay, um.

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Number 5, we just did actually last time on Thursday, independent means there if the variables are independent, then the.

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Ad.

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Always the expectations.

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Okay, we saw that on Thursday and chilly.

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Um, so.

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Number 5, the variance and so be 20.

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If the uniform random variables, and they're independent, and then the expectations add always.

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Started their uniform on the intervals 0 1.

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The expected value for each of them is point 5.

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Okay.

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The maximum expected value of the maximum. Now, how would you do that? This would be an excuse to review that perhaps.

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If we think about how we do that.

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Here, man.

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It can look to distribute the C. D. F. is appropriate.

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Okay, interesting lag there. I'm just watching.

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When I was sharing the screen, it actually was delayed a little the last.

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Syllable took a 2nd or 2 to come up again.

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Don't ask me why okay if we think about this a little.

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And we're going to X and Y, be independent.

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Just makes life easier. Okay so.

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Think of the CDF.

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Using Lowercase because there are specific things.

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At the Max is easily, that means they're both cynical to Z.

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Now, if they're independent, um.

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Then we can split this up.

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Independent, so.

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Now, what this is here is.

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I just did something wrong here should be using up for case here.

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It should be out per case.

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Okay, and this is the.

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Um, and you're assuming they're the same and.

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Your ID then the today.

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Squared Chile.

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Example slip back to the screen. We were talking about independent uniforms. 0, 1.

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Then just remind you f of X.

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Equals this and upper case.

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8 calls this.

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I e0 X less than equal to 0 X.

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I think 1 and 1.

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If X greater than 1.

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Okay, so as of Z is the square of that is.

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Is X squared? X1 okay.

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On what we want to know is the expected value.

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And the.

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Did the cumulative is X squared for the max of the 2 variables? The.

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The density function is the derivative of that.

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As the, I'm sorry, this Tuesday.

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Um, okay.

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And then the mean.

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Is the girls the times.

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F, a. Z T. Z. it goes to the girl.

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The square 0 to 1 Z.

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Which equals 2 thirds.

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Yeah, okay. And in general, um.

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Basically, the.

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The mean of the max.

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And form and 0, 1.

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A goals add Chile and offer N plus 1. so.

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Okay, 8, we did something like that a long time ago.

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Could be worth doing it again.

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And again, so we have the experiment toss the 2 fair coins. Assume they're independent. Can I show the last page?

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This 1, or do you want the previous page here?

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That 1, unfortunately, I can't straddle the page break line. I have to show a whole number of pages, but.

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I can attempt this the 1 before this. Okay. You're doing screen grabs so okay.

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I'll try to upload it as a PDF and but feel free to do screen grabs so of it.

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Let me know when you're done, so okay.

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Okay, so we're looking at question 8 toss and again to the tossed the 2 coins.

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You can define whatever random variables you like. It depends what you want to observe.

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So, for looking at the expected number of heads.

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Well.

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Basically, 4 outcomes.

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And their equal probability.

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And had kale.

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Cal had tale tail and so the expected number of heads.

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Is going to be 1 so now you might ask yourself why are those 4 possibilities all equal probability?

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You sort of convince yourself by saying you toss the 1st coins.

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You might that's a good question.

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Silence.

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Well, you know, I don't know, you could say draw might say, draw a tree.

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1st toss.

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1, half and half perhaps and then here goes the 2nd toss.

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1, half and a half and a half and a half.

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And tail tail, so if you look at that, and then the final output here is going to be like.

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And head was a quarter and kale was a quarter.

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Yeah, and once a quarter and kale tail was a quarter and so on. Okay. Exception number of heads is 1.

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So now.

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Why so I'm going to define another random variable costs when the 1st head occurred.

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Um, and I want to know.

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Oh, sorry for the proofs. And the probability that X equals 1, well, is going to be a half. So.

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And see down here, okay, the probability that Y equals 1 the toss when the 1st had occurred.

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Well, again, and something as simple like this, I look this look at my you might say my event tree.

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Um.

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So, why is the 1st head occurred here? It occurred on the 1st toss here to cut to the 1st task on the 2nd toss. And here, it did not occur at all, which will defined by 0.

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So, the probability that Y equals 1 equals 1 half you say.

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Cause on 2 tosses of the 42 outcomes from the 4.

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It could happen on the 1st time so.

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Let me try something here. What's the 2nd.

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I haven't figured out how to cut out those black boundaries on the left. Oh, okay. Good.

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Now, on.

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The probability and X equals 1 and why equals 1.

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X was a number of heads.

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Well, if I can squeeze stuff into the margin here.

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Exit calls to 1 1. 0.

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So the probability.

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It's 1 quarter, because there's only 1 of the 4 so.

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Okay, now we're starting to get a little weird or probably Y equals. Okay. The Y equals 1 given X equals 1.

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Again, squeeze it into the corner here.

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Well, again, here you just look at the things. Why was 1 given.

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X equals 1 are the 2nd and 3rd.

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For why was 1 of them? Like, was 1 1.

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The probability X equals 1.

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Given Y, equals 1 equals 1 on the 1st and 2nd and 1 of them. And so in this particular case, you can just look at that. You could also.

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You know, obviously do it in more detail for example.

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Silence.

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And get the same thing come on flip there you go.

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Okay, okay so the normal distribution.

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Then it's going to be something like this.

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40,600.

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Let me clean this up a little more.

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Something like that um, and if we go to the table.

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Which I actually had last class or something.

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Yeah, so basically, um, and I could take it down to 200 or something. Let me see here.

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So this thing here is about 1% in that area from minus 2 to minus 3 it's 1, half percent.

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Actually, even my good gross numbers.

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It's give or take 5% in this area.

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I'm sorry, I'm off by 1.

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No, it's okay because this here is the mean.

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This is a mean minus segment is semi minus 2 Sigma. This is a mean minus 3 Sigma. So below minus 3. so if I'm minus 3 to minus 2.

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Give or take 5% for minus 2 to minus 1.

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Is I'm looking at them? Sorry my mind is going on.

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Below minus 3 is point 1 per cent.

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Minus free to minus 2 are looking at differences in the big f column is.

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Say, 2% from minus 2 to minus 1 is 2% up to 15.

200
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It would take 13% from minus 1.

201
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To 0 is.

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Is like 34%.

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And then this will be again, you know.

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These are very rough numbers and so on.

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2%.

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I ended up say, point 1.

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Okay, so the question is.

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Where do we go here? So probably the particular score is from 4 to 600 that's minus minus.

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Plus Sigma. Let me actually write these different things here. So this is.

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Saying that.

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So, from 4 to 6, that's going to be about 68% give or take. I'm not going to quibble about the last digit.

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This a 2nd, here.

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Okay, I got a chat window open my 2nd, I got 2 laptops in an iPad in front of me.

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Okay, I actually have better hardware available when I'm sitting in my home office and I have, and I would have if I'm standing in a classroom at our Pi.

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Probably faster Internet also. Okay.

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That was 13 question 14.

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Okay, so now this is getting number 14 is getting into questions of samples so the original distribution, the signal is 100.

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If I take a sample of 4 students, then the mean of the sample.

219
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As a standard deviation that goes down by square root of an.

220
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So, it may be, let me write that down.

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So, we're talking here about Co rushed and for.

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Team, so the.

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Okay, so the population, so you can make those 100.

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So, we're going to take a sample of 4 students.

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Um, the, the Sigma.

226
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Of the sample mean, it's going to be 100 over square root of 4 equals 50.

227
00:33:04.499 --> 00:33:10.378
So this is the mean of the sample does not jump around as much as 1.

228
00:33:10.378 --> 00:33:21.808
As 1 student, so again, the thing is that the mean or the sample starts is an Estimator of the mean of the original population and the bigger the sample.

229
00:33:21.808 --> 00:33:30.209
Then what the tighter it is elastic and bounce around. So, let me maybe write that down.

230
00:33:54.148 --> 00:33:57.929
But, you know, it bounces around. Okay.

231
00:33:57.929 --> 00:34:04.378
You know, different samples of different means.

232
00:34:17.188 --> 00:34:20.278
Okay.

233
00:34:22.708 --> 00:34:26.789
Larger samples.

234
00:34:26.789 --> 00:34:30.179
Bounce less. Okay.

235
00:34:31.259 --> 00:34:36.568
Um, and.

236
00:34:36.568 --> 00:34:39.929
So, the Sigma of the mean of the sample.

237
00:34:39.929 --> 00:34:44.728
Is 50 in this case okay.

238
00:34:44.728 --> 00:34:48.119
So, 400 to 600 for the sample.

239
00:34:48.119 --> 00:34:52.469
That's 2 segments all sides. So, um.

240
00:35:02.458 --> 00:35:06.478
That's, um.

241
00:35:08.248 --> 00:35:12.028
That's 2 Sigma is.

242
00:35:13.679 --> 00:35:28.259
That's 2 segments each way. And the probability if I go back to my thing on the previous page is basically 96% give or take.

243
00:35:30.389 --> 00:35:35.608
So.

244
00:35:36.869 --> 00:35:46.228
And again, this whole thing with sample means gets into when you have estimation polls, full takers.

245
00:35:46.228 --> 00:35:58.378
You know, they say something that we think the vote in the election will be. The Tartan party gets whatever within.

246
00:35:58.378 --> 00:36:02.369
60 to 70% of the boats, you know.

247
00:36:02.369 --> 00:36:05.398
95 times out of a 100 or something.

248
00:36:09.568 --> 00:36:19.679
9 students, then the Sigma of the mean, as a sample mean is going to be 33. so this is 3 segments. Both ways.

249
00:36:19.679 --> 00:36:23.759
And it's whatever, I don't know, probably E, or something like that.

250
00:36:23.759 --> 00:36:27.298
And for the for soon sample.

251
00:36:28.528 --> 00:36:37.318
I should have said 16 is worded. Bad should be the standard deviation of the mean of the sample and 59. it's 33.

252
00:36:39.449 --> 00:36:43.949
Okay, questions about that.

253
00:36:45.688 --> 00:36:51.028
Okay, um, what I'd like to do is some enrichment stuff.

254
00:36:52.199 --> 00:36:58.199
I want to hit you on some oh, you know, some big things that.

255
00:36:59.938 --> 00:37:06.688
That we haven't covered incredibly in detail or into book go here.

256
00:37:14.938 --> 00:37:19.259
Well, just a 2nd questions, how did I get 50?

257
00:37:19.259 --> 00:37:23.878
14 and 15. sure. Um.

258
00:37:44.548 --> 00:37:48.358
On here.

259
00:37:54.599 --> 00:38:00.838
2nd.

260
00:38:00.838 --> 00:38:06.838
Okay.

261
00:38:06.838 --> 00:38:13.860
My my screen mirroring thing just froze was started.

262
00:38:13.860 --> 00:38:24.719
Eva, give me a minute.

263
00:38:27.420 --> 00:38:37.409
Okay.

264
00:38:39.329 --> 00:38:44.639
Get later for care. Okay. Um.

265
00:38:59.940 --> 00:39:03.210
Seeing is not.

266
00:39:04.739 --> 00:39:08.190
Just started up again.

267
00:39:21.570 --> 00:39:29.940
What happened was my play app that mirrors the iPad.

268
00:39:31.530 --> 00:39:36.480
Wasn't it froze and kill it everywhere and started again?

269
00:39:45.119 --> 00:39:53.699
Good.

270
00:40:01.530 --> 00:40:07.349
Okay, the question was how I got the 50. okay.

271
00:40:07.349 --> 00:40:14.789
Silence.

272
00:40:24.719 --> 00:40:28.860
And we're assuming 200 to 800. Oh, okay.

273
00:40:28.860 --> 00:40:34.679
Um.

274
00:40:38.369 --> 00:40:45.750
Next 1. okay. Every time we draw X1, we get a different. We see something different. Okay.

275
00:40:47.159 --> 00:40:52.139
And the expected value of X1 is going to be 500.

276
00:40:53.309 --> 00:41:04.170
The standard deviation on X1 is going to be 100. nominally. Okay, this is how the was originally designed about a 100 years ago. I believe.

277
00:41:04.170 --> 00:41:07.889
Although every few years they keep changing it to confuse people.

278
00:41:07.889 --> 00:41:16.920
Okay, now, so basically, we draw so, you know what the standard deviation means then.

279
00:41:20.159 --> 00:41:24.119
Well, you know, the standard deviation and so on, um.

280
00:41:25.440 --> 00:41:29.400
Okay, and for fun variance.

281
00:41:29.400 --> 00:41:33.300
X1, it's going to be 10,000.

282
00:41:34.409 --> 00:41:40.559
And that's the expected value of X squared minus the expected value minus.

283
00:41:40.559 --> 00:41:43.860
Square okay now.

284
00:41:46.320 --> 00:41:50.159
Let's suppose we, um, we draw.

285
00:41:51.269 --> 00:41:55.469
It's like drawing cart, we draw 4 random scores.

286
00:41:57.300 --> 00:42:00.449
X1 X4.

287
00:42:01.679 --> 00:42:05.219
And calculate.

288
00:42:05.219 --> 00:42:08.610
Y, equals to.

289
00:42:08.610 --> 00:42:18.300
Okay, this is a random so why is also.

290
00:42:18.300 --> 00:42:28.349
Random variable it has a mean and standard deviation excepted value. Why?

291
00:42:28.349 --> 00:42:31.409
It's going to be 500 still.

292
00:42:31.409 --> 00:42:35.969
I make that clear.

293
00:42:35.969 --> 00:42:40.019
And but and then the standard deviation of why.

294
00:42:41.280 --> 00:42:44.369
Is going to be the standard deviation of X.

295
00:42:44.369 --> 00:42:48.090
Divided by the square root of and.

296
00:42:48.090 --> 00:42:53.699
Which is so.

297
00:42:55.769 --> 00:43:03.809
You know, it's going to be it does not why does not bounce around as as much as X did.

298
00:43:05.460 --> 00:43:13.949
Say, 100.

299
00:43:13.949 --> 00:43:26.070
Scores okay okay. Then the expected value of Z.

300
00:43:26.070 --> 00:43:29.460
It's still going to be 500.

301
00:43:29.460 --> 00:43:32.489
But the standard deviation on Z.

302
00:43:32.489 --> 00:43:38.760
It is going to be 500 is going to be 100 the original standard deviation.

303
00:43:38.760 --> 00:43:43.679
Let me write it down in more detail.

304
00:43:48.690 --> 00:43:53.369
Okay, so.

305
00:43:53.369 --> 00:44:05.969
When we have the bigger the sample, and we take the mean. So every time I take 100 scores in average that I'm going to get a different mean.

306
00:44:07.735 --> 00:44:21.954
But these means, but these means of 100 scores, they're clustered more tightly. They don't jump around as much as single scores do. Because you take a set of 100 score is going to have some high ones that are going to have some low ones.

307
00:44:21.954 --> 00:44:23.965
And they're mostly going to balance each other out.

308
00:44:24.809 --> 00:44:29.909
Not completely, but mostly, so, I mean, so does that make sense?

309
00:44:31.590 --> 00:44:35.610
Um, so now.

310
00:44:37.710 --> 00:44:44.010
Okay, so now, so I'm getting back to 14 question 14 and so on.

311
00:44:44.010 --> 00:44:49.320
Um.

312
00:44:49.320 --> 00:44:55.500
And so X, so why.

313
00:44:55.500 --> 00:44:58.619
So, why is it I mean.

314
00:44:59.760 --> 00:45:06.360
Of the 4 scores expected value of why.

315
00:45:06.360 --> 00:45:11.909
It was 500 standard deviation of Y, equals 50.

316
00:45:11.909 --> 00:45:16.920
Okay, so 400.

317
00:45:16.920 --> 00:45:21.210
Is call that view and call that Sigma.

318
00:45:21.210 --> 00:45:24.989
So, 400 is minus 2 Sigma.

319
00:45:26.610 --> 00:45:35.699
600 plus 2 Sigma for this random variable. That's the 4. that's the mean of the 4. okay. So.

320
00:45:39.150 --> 00:45:52.650
So, the probability that if I use a notation from that table, which let me scroll down to it.

321
00:45:52.650 --> 00:45:58.469
Here.

322
00:45:58.469 --> 00:46:06.690
That's f of 2 minus half of minus 2.

323
00:46:06.690 --> 00:46:09.690
And half of 2 is.

324
00:46:12.539 --> 00:46:15.929
Point 9, 8.

325
00:46:15.929 --> 00:46:19.769
Minus point 0 2.96.

326
00:46:19.769 --> 00:46:23.940
Does that make sense? Okay.

327
00:46:23.940 --> 00:46:26.940
Um.

328
00:46:29.190 --> 00:46:32.190
So, 15, I don't need to do that and it's, you know.

329
00:46:34.889 --> 00:46:47.969
Okay, so let me look and see some. Let me just.

330
00:46:47.969 --> 00:46:55.320
At some point.

331
00:46:57.449 --> 00:47:03.630
And this is like chapter 7 sums of random variables and so on Central limits there.

332
00:47:04.074 --> 00:47:17.875
We had big stuff. We haven't done this enrichment stuff and using the textbook using Garcia for 1 semester.

333
00:47:18.144 --> 00:47:19.974
I basically did what's mentioned.

334
00:47:20.280 --> 00:47:23.730
In the introduction to the book.

335
00:47:25.320 --> 00:47:28.710
I said some stuff that I.

336
00:47:30.630 --> 00:47:34.559
I mean, you had some stuff here off and on that.

337
00:47:36.210 --> 00:47:40.199
That it's good if you see the terms here.

338
00:47:40.199 --> 00:47:45.150
It'd be a like a signal processing thing here.

339
00:47:46.199 --> 00:47:50.070
Okay, so what's happening mark off change.

340
00:47:50.070 --> 00:47:54.900
Is that your random variables? Depend on each other.

341
00:47:54.900 --> 00:47:59.280
So, here, we're assuming they're all independent, so.

342
00:48:00.570 --> 00:48:07.619
So, an example would be that you're analyzing speech, for example, or you're analyzing an image.

343
00:48:07.619 --> 00:48:19.739
The amplitude in a given millisecond. Depends depends on the previous real. 2nd, so you have effector random variables, but they're correlated.

344
00:48:19.739 --> 00:48:29.670
And they're correlated perhaps around a particular correlated to the several proceeding ones. And perhaps in some complicated fashion.

345
00:48:29.670 --> 00:48:38.070
And this is very important, for example, for image compression for speech, compression, audio, compression.

346
00:48:38.070 --> 00:48:50.639
Is that you model the correlation of a given voltage or in pixel intensity to the previous ones and the better you can model this correlation than the better you can compress.

347
00:48:50.639 --> 00:48:56.699
And that's what Mark off chains are you have you go from.

348
00:48:58.260 --> 00:49:02.369
You know, random variable to random variable.

349
00:49:04.019 --> 00:49:07.829
So, the probability of X, okay.

350
00:49:07.829 --> 00:49:14.460
Plus 1 depends on the previous ones. Okay. And that's just the I just gave you the.

351
00:49:14.460 --> 00:49:18.000
On the intellectual content there, mark off chains.

352
00:49:18.000 --> 00:49:23.789
Um, okay and.

353
00:49:24.960 --> 00:49:30.150
And again, a big application, for example, is is compression.

354
00:49:30.150 --> 00:49:43.230
Things like, this are also being used for autonomy vehicle navigation. They're trying to model streets and roads and mark off chains are deep in the mathematics of the modeling.

355
00:49:43.230 --> 00:49:46.349
I'm killing theory.

356
00:49:46.349 --> 00:49:50.940
Is you've got you again you've got packets appearing at a switch or something.

357
00:49:50.940 --> 00:49:57.690
And or, you know, your or requests hit a, um.

358
00:49:57.690 --> 00:50:01.619
A web server and the, the thing here.

359
00:50:01.619 --> 00:50:06.480
Is that the packets or requests for a page and they pile up.

360
00:50:06.480 --> 00:50:14.909
There are, they can't all be handled immediately so you'd like to model how long the queue gets you'd like to model.

361
00:50:14.909 --> 00:50:21.239
What the weight is for a packet wait W.

362
00:50:21.239 --> 00:50:33.869
The time that, before you can serve a pack before you can handle the request for service and again you get into customer quality issues. And so so a lot of mathematics has been thrown at what's called queuing theory.

363
00:50:33.869 --> 00:50:39.119
Um, that.

364
00:50:40.409 --> 00:50:45.840
Random signals again that.

365
00:50:45.840 --> 00:50:55.710
You've got your vector and does correlation power. specful density relates to statistical ways, or the random variables in the vector.

366
00:50:55.710 --> 00:51:07.349
Is How it's correlated with its predecessors. The simple thing is just a correlation coefficient. Like, we had the 2 variable calcium and so on, but you could get more complicated.

367
00:51:07.349 --> 00:51:18.809
Correlations between them and they get into things like something called specful density. If you've heard of things like white noise versus pink noise, then that.

368
00:51:18.809 --> 00:51:25.739
That relates to how a signals, a voltage correlated with its predecessors.

369
00:51:25.739 --> 00:51:33.150
Again, it gets into not just compressed and it gets into control theory.

370
00:51:33.150 --> 00:51:47.670
You're trying to have a feedback loop or something year canonical example, you're balancing a pen on your hand, and you want to move your hand back and forth. So the pen stays up, right?

371
00:51:47.670 --> 00:51:50.880
Um, so you have a filter.

372
00:51:50.880 --> 00:51:54.360
Relating to a column and filter for how you'd move your, um.

373
00:51:54.360 --> 00:52:05.610
You're looking at the angle that the pen makes, it's not completely upright and you move your hand, depending on the angle to keep it upright and no human beings can do it hardly but a computer can do it. Let's say.

374
00:52:05.610 --> 00:52:09.750
That goes into also things like.

375
00:52:09.750 --> 00:52:12.929
Flight control systems for aircraft.

376
00:52:12.929 --> 00:52:21.329
And because the complicating factor there is, you can make your aircraft really stable.

377
00:52:21.329 --> 00:52:27.300
But if your aircraft is easy to control, like, it's a fighter aircraft, then.

378
00:52:27.300 --> 00:52:37.289
It's going to be closer to being chaos. It's going to be closer to just crashing, but it's very controllable. And this gets into some of the column and filter and so on here.

379
00:52:37.289 --> 00:52:41.730
Random processes.

380
00:52:41.730 --> 00:52:49.320
Statistics and again, just to remind you statistics, if you do not know the parameter, like the mean or the Sigma distribution.

381
00:52:49.320 --> 00:52:58.619
You make observations and from then you get a confidence interval on what the true meaning of the distribution is or it may be 2 distributions.

382
00:52:58.619 --> 00:53:02.400
And you're trying to say, do they have the same.

383
00:53:02.400 --> 00:53:08.699
Mean, let's say, and the way this is worded.

384
00:53:08.815 --> 00:53:18.264
Is that if you had the 2 distributions, they can say, and we do our observations what's the probability of these 2 distributions have impact impact?

385
00:53:18.264 --> 00:53:27.144
Have the same me if they really did have the same mean what's the probability that our observed sample means would be as far apart as we observe think is again is PERRI.

386
00:53:27.144 --> 00:53:39.594
We measured the alcohol content in the 2 runs then and we observed a slight difference. But maybe the 2 runs actually had the same alcohol content. It was just chance.

387
00:53:39.869 --> 00:53:47.489
Then we saw this big a difference. Well, what's the probability we would have seen this bigger difference if, in fact, the 2 runs were the same.

388
00:53:47.489 --> 00:53:53.219
And if they were different to estimate what the difference is, so that's getting into some of that statistics there.

389
00:53:53.219 --> 00:54:00.960
We joke, like, it's statistics, some little things that I skipped.

390
00:54:00.960 --> 00:54:05.340
Were the transforms and the.

391
00:54:07.500 --> 00:54:14.280
Oh, I skipped talking about how you generate random variables and the, if you want to have.

392
00:54:14.815 --> 00:54:29.304
The quick summary, it's very difficult to do round numbers correctly. You want to try to call probably call a package if you want to go into detail, there's something called the generator. I would recommend if you want to drill down to how it's done.

393
00:54:29.514 --> 00:54:30.775
There have been some widely.

394
00:54:32.250 --> 00:54:40.050
Advertised methods in the past, something called linear, which, in fact, are quite bad. And I cited you examples of, say.

395
00:54:40.050 --> 00:54:48.780
Government authorities got it wrong. The other thing I did not mention are transforms and so on so much.

396
00:54:48.780 --> 00:54:53.130
Had to drop something out and where are they in here?

397
00:54:53.130 --> 00:54:58.590
It's like a 48 transform transform methods may be.

398
00:54:58.590 --> 00:55:09.510
Characteristic function, these are related to affiliate transforms of a time series. A characteristic function. It has a.

399
00:55:09.510 --> 00:55:13.590
Well, you you look at the definition here, you're seeing similarities, right there.

400
00:55:15.295 --> 00:55:28.824
And so you've involving to density functions PDFs you could do the same thing by multiplying the 2 characteristic functions and the involving the result. So they have some advantages here.

401
00:55:29.099 --> 00:55:33.090
So that's a big thing that I did not cover transforms.

402
00:55:33.090 --> 00:55:38.130
I maybe could've done reliability a little more. I hope I gave you an executive summary.

403
00:55:38.130 --> 00:55:52.434
Of what it is well, since the virus covert has been in the news so much of last year you could talk about things in a lifetime. What's the given that you're at? Age 40?

404
00:55:52.675 --> 00:56:06.204
What's the what's your expected? Lifetime your age 20? So, what you might say, what's your reliability given that you're still alive at age 20 or 30 or 40 or 50 or what's your probability of dying in the next day?

405
00:56:06.510 --> 00:56:12.179
This turns out to be exponential. It actually doubles with every 8 years. You get older or something.

406
00:56:12.179 --> 00:56:23.760
Um, and obviously an engineering, it's important because you build you got to pay to put redundancy in the system and obviously insurance companies are interested.

407
00:56:23.760 --> 00:56:28.889
Okay, so liability I might have done a touch more on.

408
00:56:29.125 --> 00:56:40.675
And but, as I said, I couldn't do anything and also, honestly, this year I was going a little more likely than I would have, say, a year or 22 years ago. Perhaps a year ago we're already and the problem.

409
00:56:40.675 --> 00:56:45.985
But 2 years ago, I think I might have been a little tougher on you guys and I was this year.

410
00:56:46.500 --> 00:56:53.610
And the okay, the other thing that.

411
00:56:53.610 --> 00:57:01.980
I go back to my blog, there are a couple of pages are very important blog blog blog blog. Where is it?

412
00:57:01.980 --> 00:57:07.949
Here we go. Oops.

413
00:57:07.949 --> 00:57:16.230
I wrote down some page numbers if I can find them.

414
00:57:16.230 --> 00:57:30.570
On another thing, there are many special cases and again, I gave you a sampling of the special. Okay. There's I wrote a long thing and I process I wrote this 1 up myself. It might help you understand hypothesis testing.

415
00:57:30.570 --> 00:57:36.030
Again, using tools like Mathematica.

416
00:57:36.030 --> 00:57:39.750
I think are useful.

417
00:57:39.750 --> 00:57:45.090
It's enrichment I've been arguing actually.

418
00:57:45.090 --> 00:57:50.880
Inside the engineering school that we should work Mathematica into the curriculum more. I haven't won yet.

419
00:57:50.880 --> 00:57:54.960
But, give me time.

420
00:57:54.960 --> 00:58:03.000
Okay, no, no, you should have the say to the, I'm trying to find where is the page numbers here.

421
00:58:04.590 --> 00:58:11.190
Okay, well, I don't want to waste your time too much. There's a page of discrete probability distributions and.

422
00:58:11.190 --> 00:58:14.219
And another page of continuous ones.

423
00:58:14.219 --> 00:58:26.159
So, there's many special purpose other distributions I gave you a sampling of them Chi squared, gamma, whatever and each of them has their has to use.

424
00:58:26.159 --> 00:58:33.719
And so I'm assuming your smart people. Yeah. And if you need to use Chi squared, let's say.

425
00:58:33.719 --> 00:58:38.909
You can look it up example of Chi squared would be.

426
00:58:38.909 --> 00:58:46.409
Suppose they've got a hypothesis that your birth dates are equally distributed throughout here at 12 of your birth dates occur in each month.

427
00:58:46.409 --> 00:58:50.730
And now suppose I took the class and which says give or take 80 students.

428
00:58:50.730 --> 00:59:00.900
And I counted how many people were born in January how many people in February, and so on slide 12 actual frequency accounts for your students in the class and.

429
00:59:00.900 --> 00:59:04.559
And they're not going to be exactly 8 and 80 divided by 12.

430
00:59:04.559 --> 00:59:15.804
For every month, for 1 thing, that's a fraction. It's going to be jumping up and down a little but then the no hypothesis would be that. In fact, the probability of of a particular person being born in any month is 112.

431
00:59:15.804 --> 00:59:23.875
so, if none of my process is true, what's the probability of me saying a distribution that's as far off from? Even as in fact, I saw.

432
00:59:24.659 --> 00:59:29.940
The Chi squared distribution would do that for you. So different distributions have different applications.

433
00:59:29.940 --> 00:59:33.480
I can't find the page in the book, which has this.

434
00:59:33.480 --> 00:59:38.099
But in any case.

435
00:59:58.050 --> 01:00:03.389
And presented to you, so, professor Rad, he has beautiful videos that I.

436
01:00:03.389 --> 01:00:11.369
I point you to them, Leon Garcia, the price of the book is obscene, but it does it is a standard book that.

437
01:00:12.054 --> 01:00:26.635
Most everyone uses, because its explanations are tolerable. They're good. And it has a lot of questions and exercises in it. I point you to that. I point you to all this stuff on the web stuff that I think is fun. Real World examples of people, you know.

438
01:00:26.909 --> 01:00:33.719
On the lotteries and, you know, I think they're cool they're live and things up. So that's my job is I see to point you to stuff.

439
01:00:33.719 --> 01:00:37.170
Um, that I think, is it so, I hope you got your, um.

440
01:00:37.170 --> 01:00:44.760
Value for the tuition, you paid for the course I hope you'll learn stuff that will help you in future courses.

441
01:00:44.760 --> 01:00:48.179
And if you like that, if you think.

442
01:00:48.684 --> 01:01:03.594
You got your money's worth from me then, and fill out the core survey and say so in practice about a quarter of the students in the class will fill out the survey. So, if we could get that number up, it helps our Pi, the department head.

443
01:01:03.594 --> 01:01:11.635
My boss sees the surveys for everyone in the department and exceptional things. Either way. It works the way up to the dean and so on.

444
01:01:11.940 --> 01:01:23.309
And really exceptional ones work that we up towards whatever. And the final thing is, I'm a prop. I enjoy talking and giving advice. So I'm available to you basically.

445
01:01:23.309 --> 01:01:32.789
Any time, as long as, you know, both of us are still alive for for any question that is legal and ethical and.

446
01:01:32.789 --> 01:01:47.489
You know, after you leave for other courses that are after you leave RPI, drop me an email. If I don't answer, you know, send it again. And I'm glad to listen to what you're doing even after you graduate and.

447
01:01:47.489 --> 01:01:51.179
Be glad to talk to you give you whatever advice I can.

448
01:01:51.179 --> 01:02:01.260
And as I said, what I owe you, our estimated letter, guaranteed letter grades if you choose the right define on it, will it be.

449
01:02:01.795 --> 01:02:13.224
I guess 3 hours, because that's what the finals typically are open book open local computer, but no communication with people or sites outside your local computer and so on.

450
01:02:13.554 --> 01:02:16.974
So you have your book online and so on you're welcome to look at that.

451
01:02:18.300 --> 01:02:25.079
The great and we will try and the format will be the same. It'll be on great scope. Multiple choice.

452
01:02:25.079 --> 01:02:33.659
And we'll try to get the grades out within 2 days, because the registrar wants me to get wants all the props to get the grades. So them fast. So.

453
01:02:33.659 --> 01:02:39.750
Okay, that is the end of the class.

454
01:02:40.585 --> 01:02:53.605
I'll stay around if there's questions you can also email questions or Webex them email. I possibly read a little faster because Webex takes a conscious effort to sign into it and see if anyone posted anything.

455
01:02:53.875 --> 01:02:56.514
I don't have a push thing running. I have to go and pull it.

456
01:02:59.219 --> 01:03:08.034
Anything well, I'm glad you like, and also questions I'm sorry for the technical issues with Webex. I don't even know what they did it.

457
01:03:08.514 --> 01:03:18.684
It was working, then it stopped my guess is their service got overloaded if you have other classes that had better solutions I mean, I'm told that.

458
01:03:18.960 --> 01:03:22.889
Other things also techniques.

459
01:03:22.889 --> 01:03:32.219
Also have problems, maybe that's wrong, but I'm open to advice on how to produce the course better next year because we like to.

460
01:03:32.219 --> 01:03:47.005
You know, we like to do quality work and the other thing is the process we're invested in, you guy's success. We see if we see years later that 1 of you gets in the newspaper in a positive sense, founded a company.

461
01:03:47.875 --> 01:04:00.414
Then we think that we did something right? So we're invested in, you're succeeding and when you can get back on campus, 1 of these centuries, go back into the Darren communication center, and look on the Windows.

462
01:04:01.045 --> 01:04:04.195
We got pictures of famous alumni.

463
01:04:04.679 --> 01:04:08.610
That could be you in 20 years. Ok, hold that thought.

464
01:04:11.099 --> 01:04:21.269
Oh, question exam. Yeah, we'll recycle a lot of homework questions and so on it'll be weighted towards the last month of the course.

465
01:04:21.269 --> 01:04:26.969
But will also include reviews you can't forget the stuff yet.

466
01:04:30.690 --> 01:04:34.019
No, other questions and.

467
01:04:52.920 --> 01:04:59.280
Huh.

468
01:05:15.360 --> 01:05:20.250
Okay.