 A lot of the understanding of statistics really began in actuary science or trying to figure out how to run insurance and make money. In fact, a lot of mathematics comes out of how to make money. So I threw in a few slides about that. I thought this was interesting because talking about the witnesses to the JFK assassination were bad insurance risks because they all ended up dead. This is a guy known for Gonzo journalism from local Kentucky named Hunter Thompson was over the top. And he wrote a pretty free style. That was one of his commentaries. He wrote about gambling. He lived on the edge and asked questions about risk. And risk is a big part of what drives study of statistics and probability. Risk assessment and risk tolerance or risk aversion. True gambling in my opinion is when there's no real assessment of risk. I thought this was a nice little anonymous poem. There was a movie called the man who never was from years ago that sort of fit it. And talking about taking risks. There was a saying that's been attributed to a lot of people but was pinned down to Frank Scully. I thought go out on the limb the fruit's out at the end. So take some chances or you don't get anywhere. And I found a point about Percules and his servant like his playing a dice. Percules for all his strength could easily lose a throw at the dice. Because fortune doesn't necessarily reflect any one thing even strength. And there's a lot of someone's yawning already. Legend about gambling and like the dead man's hand aces and eights that were dealt while Bill Hickock before he was shot in the back. And there's a lot of criminality associated with gambling. Some people that go into that often have a sketchy quality. You might not believe that. I found this it was kind of interesting kind of going into my regular time here. I need to hurry but I thought cool sign only in Texas liquor gambling smokes and God in one location. So this was just a collage and now I'm going to launch. This is what I wanted to cover in this talk and I'll let you read it. But I really want to emphasize a few points. I want to help you think about risk versus security. Certainty versus uncertainty and how you can deal with that. How you can take limited information and come to a pretty good conclusion about where the center of all that is what the thing you're looking for really is. Binomial theorem is a really, really big part of mathematics. And cobinatorics of course. And I'm going to talk about that. And binomial theorem leads to the normal distribution. I want you to end up associating the two. The normal distribution is born out of the binomial distribution. Also want to emphasize the idea of expectation. Now that was kind of confusing and used intuitively for a long time. But expectation is a generalized mathematical concept. And what we would call the first moment expectation is the mean of a distribution. And then got a couple of cool theorems by Russians leading to an easy proof of Jacob Bernoulli's law of large numbers. The wheat version is what I'm going to talk about. And I'm going to talk a little bit about Gaussian distributions. Statistics I think is interestingly duplicitous or two-faced or stands on two legs or any way you want to look at it. I think that the Roman god Janus represents it pretty well. You have the old face looking into the past and the young face looking into the future. And I think of the old face looking into the past to serve statistics and the young face looking into the future is the probability of potential opportunities. Also the other aspect of probability statistics is that it combines computational mathematics and theoretical mathematics in a way that's found really nowhere else in mathematics that I can think of. So just a quick listing of some definitions that you can look at and what statistics is. One of the important aspects of statistics and you take statistical samples and try to figure out what's going on with a larger population. You try to take them randomly. Or stochastic sampling and it's economical. It's the idea that's to save money. You can't study the entire population of anything unless it's very limited with efficiency. So by taking a sample and having tools to analyze it, you do it efficiently and effectively without using all your resources. That's how you can learn about a large universe with just a grain of sand. This little quote at the end by Stalin I wanted to counter by showing a picture of Komolkov who was an outstanding mathematician of the 20th century. He put statistics on a rigorous basis. Really brilliant guy based in Moscow. Most of his career I think. So we talk about populations and populations. We're talking about something that is out there. We're just taking little bits of it. Atoms of a gas or a scoop of grain or whatever small part of a hole that you want to study. You have some defined procedure. One of the aspects of statistics that came out around 1950 and on from Fischer whose name you might have heard was a defined algorithmic approach to experimentation. And any more with interdisciplinary approaches to experiments. Statisticians are a good thing to have on board. So I mentioned you want to have economical thoughts about this. You usually might have grants or a certain budget you have to follow. So you have to have samples that are subsets of manageable sizes. I want to talk real quickly about some numbers. One is Pi. Where does Pi come from? Well you have similar geometric figures. You have S1 is a small arc and S2 is a larger arc centered at the crossing of these two lines. And you can create a proportionality. Well the same proportionality really will work for the entire circle. The entire circle is the full arc all the way around 360 degrees or whatever. And so the circumference of the circle to the radius is a constant. And that constant is what we call Pi. Now let me see here. Let's see. I'll just say you can estimate Pi. You can take chords which are if you look at the arc between for that C1 it's a short little arc. And the chord between the two points where the radii hit the arc make a short line segment which is very close in size to the arc. The smaller the arc the more similar it is in size to the chord. So if you subdivide a circle into n segments like this then you can figure the length of the chords. There's an easy way to do that you know. You have side angle side determines a triangle and the squares of the radii two times the radius squared times the cosine of the angle between them is equal to the chord. You can actually do computational methods to estimate Pi. That's another thing I want to encourage people to do in this but before I talk about that I want to encourage people of any age get a compass get a straight edge and see what you can do. Our comedies was at least 76 when he was slaughtered by Romans and he was drawing figures in the sand. It's not just for kids and you kind of learn things by doing this. If you take a circle there's one thing I think is really curious is that you have this area of a circle has a constant as well that same constant is used for the circumference of the circle. How did this happen? On the surface of it that doesn't necessarily have to happen but if you take a circle like I had on this table top and cut sectors bisect them and align them in opposite ways you can create a finer and finer subdivision that becomes more squared or rectangular and if you look at the lower right there each side is the length of the radius and the little arcs on each end total half the circumference so that if well by one side being related to this conference you can see that Pi would be related to the area of the circle that's being geometrically squared when you take that to a limit. You'll be cutting molecules eventually with your scissors but Kepler was the first as far as I understand who did that. There's a famous problem I want to talk about that leads into some of this and this was Bernoulli's problem of compound interest just to review this real quickly if you have whatever your interest is that you're making on an investment you add it to one and multiply it times your basis and you'll get however much the amount is going to be or grow to by whatever the interest period is. If you invest your money wisely put one euro in a bank and get 3% interest all you have to do is wait until June 13th of the 23rd year if you put it in on January 1st and your euro will become two euros so what are you going to do with all that money? In any case this equation here of one plus the interest to the nth power is what we're interested in. Bernoulli asked if you have for whatever period the principal double what happens if you compounded a half period or a fourth period and you can come up with the question what happens if you did that an infinite number times you start to look at the limit of one plus one over n to the nth power and if you go to Google and you type graph and you don't have to type four but it will state what the graph is at the top you type in the search line graph and the equation you don't have to write y or anything but some equation of x and it wants to have it in the terms of x it will give you a graph now for this particular equation it's sort of problematic as you look at the lower right hand corner there it kind of fluctuates over the place but if you you can crunch with your numbers in Google and have it calculate for 10,000 or a million or whatever and it will give you numbers closer and closer to this long number which I think isn't so hard to remember 2.7, 1828, 1828 and 45, 90, 45, 23 kind of 45, 90 is one it's double the other and then back to 45 and 23 is about half 45 so I find it easy to remember from that point and that's as far as I want to bother you might want to internalize this this numbers is important in mathematics is pi and you know someone mentioned I have difficulty with the name of the mathematician in India who wrote about pi and he estimated pi as a square root of 10 somebody popped in my head one day this e is approximated pretty nicely within one percent or so by the square root the cube root of 20 that's kind of a fun fact if you're on a desert island or looking sitting in a prison cell or something any rate there's a way of doing this when you use the power of your laptop for number crunching or graphing if you change this around this limit doesn't care if n is going toward infinity or zero it cares if that argument of one over n or n is getting larger or smaller so you can switch this around and substitute one over x for n and you get this you can take the limit of one plus x to the one x exponent and if you graph that it's a little easier you change the limit it goes toward as it goes toward zero and looking at the graph now this is a this is computational mathematics this is not you know which everything just about comes down to in some way but it's a little different in a way of expressing or discovering what e is about but if you look at it there where it crosses the y-axis at x equals zero of course one over zero doesn't exist but the limit does and that's e the number e named for oiler now I don't want to I I figured this is largely a group of people that involved in education and teaching and for mathematics much of what people learn is taught to them by maybe non-mathematicians scientists people with mathematical skills but originally I learned a significant part of statistics from a chemistry professor that's not where you want to stop but and if you do that you often will end up with bits and pieces and need to work at putting putting it together but anyway I wanted to write this out for a couple reasons I wanted to let you look at it just these are really critical concepts the being fluent or comfortable with exponents and logarithms in the mathematics of involving statistics is is critical and I want to point out the cool thing about these kinds of equations if you're dealing with functions you're trying to do some kind of functional analysis and you're dealing with functions that are multiplied together and you want to separate out those variables you may be a one function of x and another for y and x and y are independent if you take the logarithm or a logarithm of the multiple it becomes the sum of the logarithms and you can do an inverse transform back to and take the exponential function of that get it back into it's a something like its original form if you need to so it converts multiplication to addition and that helps you pry apart problems once you identify what you're trying to figure out about a system and set up a problem or model to create a an equation to derive you can often use this to derive it I wanted also to talk a little bit about the exponent or the exponential function and its derivative if you look at this the important thing I want to emphasize here is that the exponential function of whatever base is always equal to itself times a limit and that limits kind of curious interesting let's look a little further you can use graphing functions to estimate limits on this sort of thing as well and in this I'm doing this is an important point really doing a function with exponent the base is e Euler's number and at zero that is a I got a touch screen here at zero it's equal to one that limits equal to one so the derivative of the exponential functions equal to the exponential function it is unique in that way it's extraordinarily useful in that way for one thing there are tricks like you develop moments what are called moments and dealing with statistical functions where you take the derivative of e to the or exponent of tx and with respect to t and that will be and then set it equal to set t equal to zero for the derivative and that will give you x it's something I want to mention it's more than I can get into it in depth in this discussion but I wanted you to be aware of moments and I'm going to mention it again later let's look at the logarithm derivative and this will come full circle here a little bit if you take the logarithm derivative you come up with the log the derivative of log is of any base is one over x times so that's that's important and remember of course and times a limit keep touching the screen the limit here should look familiar to you a little bit ago we did a limit of one plus x to the x and it was equal to e that basically is e so you have the logarithm of whatever base to the number e of the number e or just 2.7 1828 1828 45 90 45 23 okay if that happens to be to the base e e to the exponent or the logarithm of e to the base e is one so you end up with that limit becoming inconsequential and taking the derivative so this is one reason why we use exponential functions because the derivative of the exponential functions equal to itself and we use natural logs because the derivative natural logs is x is one over x and of course they're inverse functions just like when you have um linear equations y equals mx and the slope of x equals something of y is one over m the slope of the of the reciprocal of the natural log is related to the exponential function slope or or derivative in other words the derivative of an inverse function is the reciprocal of it okay here's a picture of Euler I have a collection of some of these lithographs I took pictures of and Euler was blind in one eye he spent most of his career in St. Petersburg and he was probably the most prolific mathematician in history but he was not so rigorous if you do an application of binomial theorem which I'm going to talk about in a minute to this log to this x this one plus delta x over x to the x over delta x you will come up with this expression of I keep hitting the screen sorry this expression of E in an infinite series you can also get that by simply deriving the McLaren series for exponent of x and setting x equal to zero I thought this might be helpful for a couple reasons one it emphasizes how you can explore some of these functions you might have forgotten that you have heard of before or that you wonder about by going to google and type in log x comma natural log of x comma and x minus one and x and the green line is x it's color coded the orange is x minus one and the natural log is the red line that I should use a pen or something to meet with I want to emphasize a couple things I know you can't see my pen but anyway the upper left hand corner on the graph is a way of expanding the whole graph you click on plus or minus above and below the cross hairs there or to the side there you click on it it'll show you vertical horizontal axis separately and let you expand either axis and I think for teaching or just use to explore mathematics for any age this is really incredible to me I don't want to see how long I've been doing this but I'll say that people like Euler he was able to do these series in his head to go out a great many terms to get an estimate he could look at a thing for a moment and know if it was diverging or converging and now you can just do it in the computer so you've been Eulerized one other point besides encouragement to use this graph function in Google which I just happened to come across and I hadn't come across anyone else that knew about it so I wanted to emphasize it to you folks is if you look at the red curve which is the natural log and the orange curve which is x minus one so I just placed the green line over by one it's at a tangent there and the natural log the derivative of the natural log is one over x and a one one over x is one so the slope to a tangent of the curve that x equals one should be or a straight line you know approximated at that point should have the slope one you see it does I think it's really important to um I'm tapping I'm sorry and I would I hit my computer with using a pen with it and tapped on the screen apologize for the noise at any rate I think getting a sense of real life experience with what happens with functions you can't break these suckers you can do whatever you want to them you won't increase the carbon footprint you won't hurt the world you won't kill animals no animals were injured or harmed and they're making this presentation now you can do all this stuff pretty safely it's probably quite economical for your own budget as well so any rate here again I just wanted to show these inverse functions now I better pick up speed this is Pascal he's on my wall in my dining room it's one of my collection he was something of a father to probability theory in the early points of it and here's Pascal's triangle which actually goes back to ancient times at least ancient persia if you look at this triangle any number is equal to the sum of the two numbers just above it and what I would encourage you look at each row as the first row is zero the second row is one the third the third row will call two and so forth you number the rows from zero and you number the items zero and one for the row one number across from zero up and number down from zero up if that makes sense I don't you can't see my cursor on my computer can you I'm sure you can't any rate of what you imagine if well take the number four there that's for the fifth that's for n equals five and that's for k equals two k being this I'm sorry k equals one so it's the fifth row first term one is the zero term this relates to compensatorics and so the four is basically would be written five items combined oh let's see if my signal is right you have zero one I'm just taking k times combined in k and items combined in k ways k at a time okay zero one okay five items combined one at a time I think would be five I'm going to go on I'm getting myself confused my point though was that if you are looking for the combinations of anything it would be all the combinations taken at the the combinations have been things taken k times without concern for order would be the same as the combinations for n minus one times taken k at a time plus the n minus one items taken k minus one at a time because each of those ones that are taken k minus one at a time get get a new one wouldn't you go to the next row anyway I've twisted my tongue here somehow and I'm going to go on though because I've got to cover some stuff here Pascal's triangle I took excel and graphed it and this is a I think an important slide for you if you look at this does anyone notice anything about the shape of that curve uh plotting out uh Pascal's triangle as those numbers get larger those curves become more and more like a normal bell shaped curve Pascal's triangle is the spine of the normal distribution in a sense there was a Abraham de Moivre who uh was the first to recognize that a normal distribution could be used to approximate a binomial distribution especially when you get into larger numbers and uh it's tedious to work with binomial distributions so if you get into because you look at the coefficients uh I had meant to talk about in factorial here in factorial is equal to itself times every number less than itself down to one and zero factorial and one factorial or one those get to be really large numbers to deal with if you're working with pen and pencil that's going to be a lot of work and take down a lot of trees so using a technique for approximating binomial distribution with normal distributions becomes quite useful if you need to do that um also just I wanted to emphasize this again in functional analysis if you look at take the logarithm of in factorial you break that product up into sums and if you consider instead of n you kind of imagine it on a a graph instead of n think of x and um do it as though you're uh doing an integral of the logarithm of x that's solvable and that is a little direction for you and how you can derive what's called Stirling's approximation again de Moivre uh was the first to come up with that although uh Stirling improved on the constant and he was English so it was named after him de Moivre was quite interesting he was a Huguenot he had to flee he was put in prison for a while he never married he was a bright guy he had to leave France because of the persecution of the Huguenots and the Protestants French Protestants and he went to no de Moivre was French and he went to London and he studied all the time he did tutoring of wealthy people someone actually gave him Principia that Newton had published and he would take pages tear pages out and read them as he was going to his next job tutoring and I heard that it just broke my heart to think that the book being destroyed but that was the only way he would study it page by page and he found it to be much deeper than the other books he had read uh in any case there was a story also that Newton had people come up to him much the way Brahms would have people come up and want to show their scores that they'd come up with and he would see musical cliches in them just by looking at the score Newton would have people come to him at all kinds of times wanting to look at some problem that they had worked on and he was supposedly quoted as saying might want to talk to de Moivre about that he knows more about these things than I do so this is a graph of the sterling equation where I put x instead of n just to show how fast that grows okay and if you change the scale this is for the x-axis spread out pretty wide so you're seeing just in factorial estimated at the low numbers if you go to a higher scale look at that caveman go he flies all right permutations I want to just point out one thing about permutations and you can look at this kind of a graph you probably or table you've probably seen before about how you get a formula for permutations have been things taken k times that is related to the binomial theorem this is the coefficient for binomial theorem and if you look at the factorial over n minus one factorial there that's of course you have many more permutations than combinations because combinations don't care about order and and permutations are ordered sets so how many more k factorial so I thought that was rather interesting and Newton during the plague came up with a generalized binomial theorem which I wrote there at the bottom which has these equations but at any rate if you expand the permutation thing and Newton's Newton's numerator is equal to this version of the permutation number and the networks for hmm here we are that works for using binomial theorem to do almost any kind of rational exponent okay so I don't want to dwell on this too much here just showing the expansion of the binomial theorem this is a hubachian lithograph from about 1729 that's on my dining room wall that I share with you that's Isaac Newton of course okay the few definitions experiment you make observations one and only one outcome and when you do an experiment your um taking samples from uh that correspond to points in a sample or outcome space and there are different ways of talking about this as I wanted to point out also this uh the two dots and equal sign is uh one way in mathematics that people use to say is equivalent to or uh defined as so it's a kind of shorthand if you ever see that discrete sample spaces are the easier to talk about there's a couple reasons for this I'll let you read this as I talk but when you have a discrete sample space all these finite points that were accountable infinity which means that you can put them in one-to-one correspondence with the natural numbers one two three and so forth events an event is any subset of a set now especially if you get into the continuous numbers uh if you talk about uh all the subsets of the real numbers you're talking about sets of cardinality that are greater than the real numbers in other words there are different levels of infinity the real numbers have an infinity that's infinitely more than the infinity of the natural numbers so you have to be careful with the definitions in terms in these or else you'll get into the same trouble that set theory got into not what like with Bertrand Russell's paradox which is equivalent to if god is all powerful can he invent or create an object that's too large for him to move so you have simple events single sample point and uh all right I threw this in just for a laugh well if you feel this way at this point sorry it stood me right here's one point I wanted to really emphasize expectation is a term that's used a lot in statistics and they talk about expectation if you're you I haven't really defined it yet mathematically but let me point out if when when it's applied to the average or mean or and there are a whole bunch of different estimates of central tendency in distributions usually you use the average that is what's called the first moment which I'll mention again there I'm not going to talk a lot about moments but it's the expectation of a sample and that's totally linear meaning that if you multiply something times that you can take the constant out of the expectation and if you are adding two random variables the expectation is equal to some of those two I think that you need to play with the stuff to really start thinking of it in terms of expectations rather than it's almost like algebra of samples I don't know I got this far down here seem to I don't know but just lights that I'm not seeing okay well I'm going to go with the flow here the weak law of large numbers was worked on supposedly for about 20 years by Jacob Bernoulli or Jacob Bernoulli and who was a Swiss mathematician and physicist and I can believe it probably took him 20 years there were a lot of points that were confusing to people about probability when this all started and he was really looking at coin toss but the idea of it is when you have a sample you get an average there's no guarantee it's going to be the average of the big picture the average of the population but as you take bigger and bigger samples or more and more samples it will start to average out you'll start to get the true mean and he actually proved it you take a bunch of independent variables you can add them together and make another independent variable like this and for some reason a whole slew of my slides disappeared so I'm not going to get into the details there's a point about variance variance is equal to the expectation of the difference between each independent variable and the mean squared it's it's it's not linear like the average is so if you have the variance of these independent variables all being sigma squared and you take one over n that one over n becomes n squared there this is terrible I wish I had the slide created to show the basis of that but it's a tricky part of this proof that but let's you end up with this equation at the lower left hand corner which is based well I'm waiting for this to res now I'm sorry this is really all over the place I'm going to just wing it here here's the idea of how the law a weak law of large numbers was can be proven you have a a theorem called Markov's inequality and it works for any any distribution and it basically states that the chance in a distribution if the chance of a value being an outlier or greater than some value a is less than whatever the average is or expectation for that sample space divided by a and that's really beautiful now that only applies to positive distributions where there are no negative data points if you apply Markov's inequality to Chebyshev's or I'm saying this odd to get Chebyshev's inequality you apply Markov's inequality to variance and you can show that the variance is that basically it puts bounds on the difference between any any x in your sample and the the mean and you take that and then you can prove the weak law of large numbers by based on a limit see if I can get back a little bit here I'm running short on time and I wanted to leave a few minutes for questions if you had them something funky happened to my slides here I don't know they I went through them last evening and they were all in order thank you day but see if I can get anything okay okay here are some maybe I can still teach another point I hope this is by the way if you saw a picture of Prince here I was a joke I was going to talk about the prince of mathematics which is Gauss yeah they were all numbered properly last evening and I looked and I thought they were there added a few at the end that were also numbered right but well I created this scatter graph using excel I just made up numbers it's like bad science and if you look at this this is like what you're trying to what you're trying to figure is is the mean and there's a normal distribution embedded in this it's like trying to measure the position of a star and there are derivations Gauss really was the first one to get a good breakdown of this and there are cool ways to derive the Gaussian distribution and we're done by Maxwell and Herschel and others in any rate a couple of points the there's symmetry radial symmetry it's if you look at any point around the origin 180 degrees from it should have the same likelihood of having a point so you also have x and y and they should be independent variables and of course the radius to any x or y point is the square root of the sum of the squares but you need to separate these out somehow now I wanted to point out if you just play with equations on on google graph like 1 over x that has a curve to it if you could switch it around somehow maybe that would work but no that's not too good for one thing it doesn't it's as a discontinuity at zero you look at this other ways of how can you get rid of this square root of x square plus y squared you need to get rid of the radical maybe in square x squared put it in 1 over x square that gets rid of the radical but that still is this hyperbolic sort of configuration and it goes to asymptotes so that's that's clearly not going to get you what you want in terms of the shape of the function and here what if you just take the exponent of x squared that flies high in air on each side pretty quickly and it doesn't graph very well well what if you do this you know with if you got to go through x equals zero and you're dealing with a reciprocals one way to deal with that is to make it an exponent to the negative x because the negative exponent makes e in the denominator and so negative x and then you also have x squared and you get a curve that looks like this so it looks quite a lot like those little ribs on the growing Pascal triangle that I mapped out on through excel if you can see that just intuitively so I'd like you to remember is that for the Gaussian distribution the exponent to the negative x squared is fundamental to its setup the other thing is if you've got x square plus y squared you want to separate them you've got exponents you look up the left hand corner here you can turn it into e to the y that's supposed to be e to the x times e to the y it's good it's not a book or here we go like this so I want you to remember that the shape of the normal distribution relates to exponent of negative x squared and here is the equation for it all worked out once you set up constraints about symmetry and how the variables act you can come up you can come up with a problem to solve which is not easy to demonstrate it's multivariate integration so it comes out to this equation but you notice you have your x minus mu which is mu is the average squared and by putting it over whatever the standard the standard deviation squared or variance there it normalizes it in other words if you take this and plot it as a distribution and figure the variance of it the variance will be one and it will be centered because every x doesn't matter what equation you start with the every x would be subtracting the mean so the mean of the transformed one is in the center it's zero and you can substitute z is equal to your variable minus the mean over your standard deviation and you'll come out with a useful equation that can be integrated to get you the area under the curve and this is the form you go you put your data in this form you go to your tables and you can extract what you need finally I've just mentioned and I'll finish the central limit theorem is really very cool it doesn't matter what kind of distribution you have you can have any distribution and do experiments and try to determine the mean and as you do more and more experiments to determine the mean the distribution of your plot of the means derived from each of those experiments is a normal distribution that that's pretty stunning really and again it doesn't matter what kind of distribution it is there's one more aspect here I wanted to suggest you try to know the empirical rule that within once it's really standard within one standard deviation of on either side of the mean 68 percent of the data lies the probability that is 95 percent that any point of data will be within two standard deviations and it's 99.7 percent within three standard deviations so 68 95 99.7 that's the empirical rule that's worth remembering so okay we're over three standard deviations from the origin here I had created an equation which was a polynomial which by setting roots equal to zero and one it it crossed the x x-axis at x equals zero and one and it had a skewed quality to it and I was pleased to make it and I normalized it I took the area under the curve and divided it divided the polynomial by it so that if you integrate that polynomial between zero and one it was equal to one and that's one of the slides that got lost here was the axioms by which probability is defined everything has a probability in your sample space of zero or greater and the total probability of all the elements or our points in your sample space is one so if you have a probability distribution curve of which if it's discrete you call it a probability mass function it should total one and I'll give one last tip here it was a great article I read recently is about David Hilbert who died in 1943 in Germany he was a famous 20th century mathematician and his 19th problem he proposed which was solved 10 years later related to you know squares if you look at variance is equal to sums of squares in this case the square of each x minus the average so it's got to be greater than zero and he wondered if polynomials that are never negative are always sums of squares of polynomials and two polynomials squared would equal the original polynomial if you know it's positive everywhere and that's happens to be true that's being used in driverless cars to create barriers for where the car should go and where it shouldn't go it stays in the negative areas then it won't hit any of those positive polynomials it's the trouble was it was too much calculating even for computer you know with pencil and paper you may think you can do a lot and you can use a slide rule and it helps you if you've ever used a slide rule if you get a calculator you can do quicker and more precise calculations probably and you got a computer laptop you can do great things but that expands your horizons and at some point you realize that you're out at sea in a canoe your laptop is like a canoe and you can have equations and problems that your laptop will never be able to get through but this these two people developed an algorithm by which it's so like fast 4a transforms they could work through this problem at any rate there's a lot of mathematics about squares and sums of squares and variance relates to that and i had someone recently asked me why wouldn't use use absolute value for variance and the reason is because it's not useful you could you can define anything any way you want and it'll come out any way you can find relations so but with variance defined as the sums of squares of the difference between each sample value and they mean it's it works and it's useful and you can do a lot of mathematics with it i'll say one other thing real quickly especially with discrete problems like tossing a coin it's another thing i'd shown in the slides which got lost that the probability if it's a fair coin if chance of hitting heads or calling it success and tails fair would be 0.5 and the probability is 0.5 well you never have you know 0.5 isn't a choice you toss a coin you get a headset that's one that's a success and payers is zero and so the mean doesn't have to be part of the sample set as a point i wanted to make and kind of related to that i thought was the idea that half the on the average human beings have one testicle or pretty close to it the average doesn't have to actually be represented above the sample so i put a pdf of my whole set of slides that should have shown at this link that people are welcome to use and i'm going to post it right now along with the article about um uh helbert i was reading something else this morning in 1934 david helbert was having a meal with a man named roost who was the nazi minister of education and so roost asked uh helbert so how are things at gertigan gertigan was the top place for mathematics in the world at the time this is 1934 and that's where gauss had been and in any case he says so how are things at gertigan now that we've eliminated the jewish influence and helbert said there is no more mathematics at gertigan so education and freedom of thought and it doesn't belong to any ethnic group and it doesn't belong to any demographic it's human and it's a treasure and it's one of the things that i think makes it is redemption for humanity which is otherwise seems a pretty sorry lot so i'll end with that any questions i apologize my slides got buggered up so badly i don't know well thank you thank you well i wish everyone a good weekend i appreciate that just it was um i think an ambitious presentation but uh i thought i could fly through it but remember binomial theorem and normal distributions are related and play with graphs say graph whatever on google search and see what happens that looks like a good probability that of having a good weekend i appreciate the opportunity to speak here i'm going to turn my microphone off i guess thank you i think you have a point