 Hi, I'm Zor. Welcome to a new Zor education. This lecture is part of a relatively comprehensive course of advanced mathematics for teenagers, which is presented on Unizor.com. It's presented as lectures and there are notes for these lectures. Notes basically play the role of textbook. So it's very beneficiary for you if you will read the notes as well as listen to this lecture. Now this particular lecture is dedicated to normal random variables and their role. Actually I'm going to talk about concrete usage of normal distribution just to illustrate how important it is for theory of probabilities. I mentioned once that normal distribution is probably the most important distribution and this illustration basically is just to illustrate it. Okay, now what I would like to start with is just let's recall what exactly the theory of probability is all about. I mean, why do we have to study theory of probabilities? Well, the most important consequence of knowing that the probability of certain event is such and such is to be able to predict certain behavior of this event or process whatever in the future. Basically to evaluate chances when and under what conditions this particular process will go this way or that way or some variable will take this value or that value and the probability actually gives you certain statistical evaluation of what exactly will happen in the future. Now the example which I would like to use in this lecture to illustrate how the normal distribution is used for this particular purpose is related to gambling. So let's assume that you're a gambler and you're a fair gambler which means that we are talking about certain random experiment when the results are either win or lose and the probability is one half in both cases. So the probability of win equals the probability of lose equals one half. So approximately in half of the cases if you repeat the same game again and again you will win and in half you lose. And let's assign the price for the winning and the losing so we will deal with numbers rather than words. So let's say winning is the house pays you one unit of some currency. Speaking about currency I mean I live in United States so I can save dollars but at the same time I understand that some other people might actually live in other countries. So why don't we use an international currency called Bitcoin. Okay this is something relatively recently appeared in the internet that's the computerized currency. Okay so you win one Bitcoin and you lose one Bitcoin with probability of one half. Now let's say you want to play 100 games. That's your intention you go to whatever casino gaming house and you are planning to play 100 games. Well what's the worst thing which can happen. Well the worst thing is you lose all 100 games right. And since you have to pay you have to pay a dollar sorry a Bitcoin every time you lose then you have to have some kind of reserve amount. So if you go to a casino and you would like to be paid to be able to pay your debt in case you lose. Well the worst thing which can happen is in 100 games you have to have 100 Bitcoins in reserve. Well but then you think look what's the probability of losing 100 times in a row. Well it's very easy to calculate. It's one half times one half etc times one half 100 times which is one over two to the 100s power which is very small number obviously. So you are thinking that hey I mean I can take my chances and I can take let's say 99 Bitcoins in reserve rather than 100 because at least one game I will win right. Well that's true but again what's the probability of winning exactly one time out of 100. It's also relatively small. Now we basically remember that winning let's say N times out of 100 is basically number of combinations of N out of 100 times the probability of every combination which is one over two to the power of 100. So this is the probability of number of win equals to N. That's what it is. Now using this let's say you would like to take with you certain amount of Bitcoins and you would like to evaluate what's my risk which I can undertake in terms of probabilities and what's the amount of money amount of Bitcoins I have to take in reserve to actually be within that risk category. So for instance if I don't care about risk which amounts to probability of one over two to the 100 I don't have to take 100 Bitcoins as a reserve amount with me. I can take 99 but maybe I can actually be relatively satisfied with slightly higher probability. I mean I understand this is a very low probability and this is a very safe kind of a betting. But at the same time the something which is the tolerance to the risk might be significantly higher. For instance you can say you know what if my probability is less than 0.025 of winning more than my reserve amount. Yeah I can probably live with this risk. That's okay. I can take the risk which is less than 0.251. And well if God forbid it happens that I will really lose more and the probability like this really will materialize. Well I can do something. I can call my friend and he can bring me money into the casino etc. I know it's inconvenience but it's such a rare case so I think I can live with this amount of risk. So you are assigning certain amount of risk. Now the question is if you would like to basically bet on the amount of your loss would have the probability greater than this one. Now if you would like to bet on this if you would like to take this particular risk how much money do I have to take. Here is how you would probably answer it if you approach like 100% statistically correctly without any approximations etc. Now the probability of let's calculate the probability of losing let's say K bitcoins. Okay now the probability of losing K bitcoins equals probability of losing let's say N games and winning M games where M plus N is equal to 100. That's total number of games which I'm playing right? And if I win M and lose N then M minus N is something which I have basically have as a result of the game. So if the result of the game is K I'm losing bitcoins. So that's basically now K since K is losing that K is assumed to be negative basically but it doesn't really matter. For any K positive or negative you can say that winning a negative amount is actually losing. So basically these two conditions which result in M is equal to 100 plus K divided by 2 and M is equal to 100 minus K divided by 2. If I'm subtracting adding and subtracting these two equations. So these are my numbers of winning and losing to have the result of 100 series of game equal to exactly K. Now what does it mean from the probabilistic standpoint? What's the probability of this? Well we know that every particular random variable which is a result of one game is a Bernoulli variable. So we have a Bernoulli variable for one game for another game for one hundredth game. Now we are adding the results. Now every one of them is equal to plus one or minus one with the probability of one half. Now their sum is the result of one hundred games. So what do we know about each one of those? So each one of those is a Bernoulli variable. What's the expectation of let's say the first one? They are identical and independent. So the first one for instance has the expectation but it has plus one with probability of one half and minus one with probability of one half which exactly equals to zero. Which is obviously normal. I mean you understand that that's how it's supposed to be. If you win with the probability of one half and lose probability of one half the same amounts plus it's negative, then obviously you have expectation of zero. Now what's the variance? The variance is a weighted average of squares of deviations from the expectation. So expectation is zero. So the deviation of this from zero is one and we are talking about square and the probability of this one half. Now the deviation of this from zero is minus one also square also with one half probability. And it's equal to this is one and this is one so it's one. And therefore standard deviation of C1 is also equal to one because this is the square root of variance. So we know everything about one particular Bernoulli variable. Now sum of 100 Bernoulli variables if you remember is binomial distribution. We were talking about this a few times before and we know the probability for sum of Bernoulli variables. The probability of this sum to be equal to K is now if you know that the sum is equal to K you have to have m winning and n losing where K is participating in this formula. So I will just use the formula from the previous lecture. It's number of combinations from 100 to m let's say 100 plus K over 2 times probability of one particular winning and one particular losing right? No I have to multiply it sorry I have to multiply it minus K. Is that right? Times well this is supposed to be times P times Q and in the corresponding powers. But since P and Q are exactly equal to one half you will have one over two to one hundredth right? I'm not sure do I have to put this in? Let me think about it. No actually I don't have to. No I don't think I have to put it in. So I have to choose any particular and then lose lose lose. I don't think I need it. I think I was right the first time. So all you have to do is to choose winning numbers because choosing winning automatically choose the losing games right? So as soon as I pick the winning numbers which is number of combinations of this from this then everything else is predetermined. I think I'm right. All right so this is the probability of our sum to be equal to K. Now so what kind of amount do I have to take with me to be able to sustain my losses in almost all cases except certain risk which I am going to accept. Let's say I'm willing to accept this amount of risk. No more than this one. So what should I do to calculate the amount of money I need? Well I have to take the probability of A to equals to minus one hundred. I lose all games which is a very small probability and calculate it. Then I have to calculate the probability of minus ninety nine and add it to this one and continue adding while this sum is less than my limit of risk which I am assigning to myself. And wherever I stopped let's say it's A to equal to minus sixty for instance. If up to minus sixty I still get this but fifty nine if I add fifty nine it goes beyond this it means that I have to take at least sixty bitcoins with me to satisfy all these cases. Right? So I basically ignore these because some of their probabilities is less than this one but starting from let's say sixty or whatever it is I don't know. I really have to provide the amount of reserve to satisfy. Right? So it's really kind of a tedious and long calculation look at this formula. So I have to calculate this formula for all K from one hundred minus one hundred minus ninety nine etc. etc. That's a very long calculation and obviously nobody wants to do this. And here exactly where normal distribution comes very very handy. And here's why. Now the sum of these is a sum of independent identically distributed random variables. And we know that this particular sum behaves as close to normally as possible if the number of these variables is relatively large. So one hundred is a relatively large number and we obviously can evaluate the degree of how close the distribution of this which is binomial distribution. How close this binomial distribution is to normal distribution. But if you remember I was actually exemplifying with a graph. With one component in this sum then with two components then with three components which looks something like this. Then with four components it looks like this etc. So more and more distribution resembles normal. And when you have one hundred that's actually a lot of components and it's very much close to a normal distribution. What kind of normal distribution? Well normal distribution is defined by two parameters expectation and variance or standard deviation. So let's just calculate expectation and standard deviation of this sum. Well the expectation... I don't need this anymore. Now the expectation of eta is equal to sum of the expectations of these. Because expectation is additive function of random variables. And since each one is zero the sum would be equal to zero. Now variance is also additive function of variances of these because they are independent. This works only for independent variables. Now the variance of each one of them is one so we have one hundred of them so the variance is one hundred. And obviously the standard deviation is square root of one hundred which is ten. So we know that a normal distribution which is distributed very much like this one. Very very close. It's a very good approximation. This normal variable has an expectation zero and standard deviation ten. Okay now we can evaluate it using the sigma limits. Now you remember that if you have... In this case it's a symmetrical relative to zero. If you have a bell curve which represents my normal distribution with this sigma. Now this is our expectation. Now if I take sigma on both sides then this probability the area... Now this sigma would be what? Minus ten and ten and plus ten. The area would be if I'm not mistaken something like zero point six five or something like this. Now if I take two sigmas at these pieces I will have something like zero point ninety five. And if I have three sigmas I will add this one as well. Then the area from here to here now this is minus twenty this is minus thirty this is twenty this is thirty. So the area from minus thirty to plus thirty would be zero point ninety nine something I don't remember. So let's just think about when do I have to pay well when I'm losing which means these guys. So let's say I'm using the rule of two sigma which means that this particular area which on the left from minus twenty. Let's evaluate its probability. Well I know that from twenty to twenty I have zero point ninety five right. Now which means that outside of this which means this little area which is going to minus infinity. And this area which goes to plus infinity has a probability one minus zero point ninety five. Now but we are interested only in losing not winning right. So we need half of it. So this is what zero point zero five divided by two which is zero point zero two five. Alright. So if you remember I was saying that I can actually ignore all the cases which had the probability less than zero point zero twenty five. That's the risk which I'm going to take. So what's the probability of losing more than twenty bitcoins. This is so what I'm saying is if I will take only twenty bitcoins as a reserve amount in all these cases to the right of the minus twenty as a result of my one hundred series game. I'm fine. I will be able to pay. I will either win or lose but no more than twenty bitcoins. And only when these cases occur when I lose more than twenty bitcoins only then I will not be able to pay. But the probability is just the area underneath the bell curve from minus infinity to minus twenty. And we have already evaluated using the rule of two sigma. So I can say my basically my point is made. I will take twenty bitcoins with me as a reserve amount and I know that I'm not risking by more than zero point zero twenty five as a probability of not being able to pay my debt. So in twenty five out of one thousand cases I will have to call my friend to bail me out. In all other cases I will I will be fine just taking only twenty bitcoins. You see the maximum I can lose is one hundred right. But now with only twenty bitcoins I can actually satisfy my debts in almost all cases except twenty five out of out of each one thousand. So the probability is in this case is my tolerance to the to the risk basically. So if my tolerance is measured in this probability then I can say that twenty bitcoins is enough for me. So my point was for this lecture how important the normal distribution is. You see I didn't really involve all these calculations I just use the approximation. And yes I understand this is the approximation and it needs its quantification if you wish. How close the binomial distribution of a sum of one hundred Bernoulli variables to a normal distribution with the same expectation and variance. This is a different story but let's just take my word that with one hundred experiments it's close enough. So this evaluation is pretty good actually. Now my understanding was that I better give you this feeling rather than exact calculations etc. The feeling of using normal variables in cases like this. And it's very useful and it shortens completely shortens all the calculations etc etc. So normal variable is very very important for this exactly reason. Well that's it for today. I suggest you to read the notes on Unisor.com for this lecture. And well that's it. Thanks very much and good luck.