 Hi, I'm Zor. Welcome to Unizor Education. We continue course of advanced mathematics for teenagers. The course is presented on unizor.com. And this lecture is part of the theory of probabilities topics. And we will talk about normal distribution, normal random variables, and especially about certain evaluation of the values of normal random variables. Using standard deviation. It's called sigma. So that's why it's sigma limits. All right. Well, let's just think about it. Why do we need the theory of probabilities just as a subject? Why do we have to research anything related to this? Well, obviously the purpose is certain level of prediction of the values of variables which we cannot say exactly that the value will be such and such. So there is always certain random process which we are talking about and random variable is something which might take this value or another value. And our purpose using the theory of probabilities is to predict that certain values will occur more often than others, something like this. So this prediction actually is the main goal of the theory of probabilities. Well, of course, this is some kind of a utilitarian approach. Any mathematical subject has its own value and purpose without any practical applications. But in this particular case, theory of probabilities actually was developed from these practical necessities to predict the values of certain random variables. Now, you also know that in many cases, random variables behaved like so-called normal random variables, distributed along certain bell curve where the middle of this bell curve is the expectation, the mean value, and the probability of the random variable to be from to is measured by this area under this bell curve. So this is a non-formal explanation of what normal variable is all about. Of course, there is a function which actually describes this particular curve and that precisely defines the normal distribution. But we are not talking about this. We are talking about intuitive understanding of what the normal random variable is. That's how it is distributed. And the area under the entire curve is equal to 1, obviously, because that's the probability of a normal random variable to take any value, and obviously it's 1, but if you would like to evaluate what's the probability of our normal variable to be in between points A and B, then we just have to calculate the area under the bell curve from this point to this point. All right, so it's very important to be able to evaluate the values of normal random variable, well, actually to evaluate, to predict the values with certain probabilities. And that's what this lecture is all about. Now, normal variables are basically defined by two parameters by the middle point, which is called mean, or expectation, and the steepness of this curve, because it can be this way or it can be this way. Now, in both cases, the area under the curve would be equal to 1 because that's the total probability, but in the case of this second curve, the values are concentrated more around the mean. In the first curve, which I drawn, the values were more spread around. So basically the steepness how this curve actually arises is characterized by standard deviation, as we know, sigma. So there are two parameters here. Now, so we have considered this particular normal variable with expectation mean value mu and standard deviation sigma. Now we would like to evaluate the probability of this normal variable to be within certain boundaries from A to B. Now, it's very convenient to actually look at the symmetry of this picture and consider that A has a position mu minus certain distance and B has a position mu plus certain distance. So these are equal to each other. So the whole bell curve is symmetrical and we usually understand that the values close to the mean value will be more often occurring. So that's why traditionally we just have the interval around the mean value, the symmetrical interval with the mean value to be in the middle of it as an evaluation segment. And we are interested in the probability of our random variable to be within this evaluation segment between mu minus d and mu plus d. So now it looks like our probability depends on two parameters, mu and d. Well, actually it doesn't depend on mu because if you shift this to the left or to the right, the probability to be within this distance d from the mu will be exactly the same. So our probability of our random variable c to be within interval of mu minus d to mu plus d actually is a function of only two parameters. It does not depend on mu, it depends obviously on d and it depends on the steepness of this curve, sigma, right? So let me simplify it even more. Let's just measure this distance d from the midpoint in terms of sigma, in terms of standard deviation. Then it will be basically a function only of one parameter of sigma. Now, what does it mean in terms of standard deviation? Well, there are actually traditionally, there are only three different segments considered when d is equal to standard deviation c, sigma, sorry, when d is equal to two sigma and when d is equal to three sigma. So these are traditionally, I mean obviously we can measure d in any kind of terms and obviously we can calculate it using certain calculus. But I'm just saying that traditionally it's probably sufficient to characterize the whole distribution with only the value of standard deviation sigma. Now considering that mu we know. So we know that for a particular random variable we know what its distribution is. We know its average, its expectation value mu and we know its standard deviation sigma. So for this particular variable we are evaluating what's the probability of this random variable to have the value within the interval mu minus sigma mu plus sigma or minus and plus two sigma or minus and plus three sigma. So these are traditional intervals which we are considering. And obviously if for some reason we know the distribution of our random variable which means we know mu and c and these numbers, these probabilities have already been calculated because it doesn't really depend anymore on any d. We just calculated for three different concrete cases. Then we can evaluate the value of the random variable, the values which random variable takes. We can evaluate with certain probabilities within these three different intervals. And let me just make you a concrete example. Let's say we already have this thing, this is mu, this is sigma, two sigma and three sigma. So it's mu minus sigma, mu minus two sigma and mu minus three sigma. And same thing to the right. One, two, three, mu plus sigma, mu plus two sigma and mu plus three sigma. So this is area from mu minus sigma to mu plus sigma. And with this particular bell curve, with this particular mu and sigma, this value can be calculated. And the value actually is a known number. So let me just give you an example. So the probability of our normal random variable to be within the interval from mu minus sigma to mu plus sigma equals to 0.6827. Now, what if I would like to say something about the values of the random variable, how they fall on this line? Well, they can fall here, they can fall here, they can fall here. Well, obviously this is area where it's more concentrated, but in theory there is a certain non-zero probability that our value will be somewhere there as well, right? So what I'm saying is that if the whole area underneath of this bell curve is equal to one, the area underneath this curve from mu minus sigma to mu plus sigma is equal to this, which means that the probability of our value, our value of the random variable to fall within these boundaries, to fall within this interval is this. Well, what does it mean? It means that statistically if we will perform our experiment, which results actually in the normal variable with these distributions mu and sigma, if we will repeat it again and again and again, then relative frequency of the value to fall inside of this interval is equal to this. So approximately 68 out of 100 cases we will fall within this area and in other cases outside. Does it mean that we can predict the value with a certainty? Well, certainty just completely out of the question, but there is some approximation. Well, we can say that 68 out of 100 approximately experiments will result in the value here. Is it a good prediction? Well, 68 out of 100, not really a good prediction. I mean, it's not really like a significant majority of the cases, but let's consider the interval which is slightly wider than this. Let's consider mu minus 2 sigma and plus 2 sigma. This is an interval. So we are adding these areas as well. Now, to be within this interval is much easier, right? I mean, it's wider, so whenever the random variable takes value, it's definitely more probable that it will be within this wider interval than within this one. Now, how much more probable? Here is the number. 9-5-45. Well, that's a significant growth, you see. So these areas bring a lot. So if it was something like 68% of the cases, now it's more than 95% of the cases when our random variable will fall within this wider interval. So now that makes our prediction significantly more precise. So if we are saying that with the probability of 95% or slightly more than 95%, our random variable will take value from this to this. This statement has much more validity. At the same time, we are widening the interval. So what's the value of our prediction? I mean, if we will predict that our value will be from minus infinity to plus infinity, the probability of this will be 1, but it doesn't really give us any information. So what I would like to say is that we are giving a less precise evaluation. We are giving a wider interval, but we are having this evaluation with a greater probability. I mean, whoever is asking us should really choose what he prefers to have a narrower, more precise evaluation of the values of the random variable with lower probability and wider evaluation with the higher probability. Well, it actually depends on the person and on a concrete task which is in front of him because if sigma is really very, very low, because if values are already concentrated within this very narrow interval around its mean, around mu, then probably we can expand the interval to 2 sigma and say that, well, the probability is very high that it will be within this interval and the precision of this interval is sufficient for a concrete task which the person performs. Well, and now let me just widen it even more. We add these areas. So we are doing from minus 3 sigma to plus 3 sigma. Mu minus 3 sigma to mu plus 3 sigma. And probability is 99.73. Wow, this is almost like 100%, right? Almost like 1, which means that these little tails are really insignificant. Not that it's not possible to get the value somewhere in this area, it is still possible with the probability of 0.0027. However, it's a very low probability, which means statistically, if we will perform our experiment again and again and again, I can say that almost everything will be within this wider interval from minus 3 sigma to plus 3 sigma around the mean value. So our prediction will be less precise, but with more probability. That's always a trade-off. So any practical problem has this kind of a trade-off. If we want to precisely evaluate the result of the experiment, then the probability of our prediction might not be very high, which means it can or maybe it will, maybe it will not. I mean the probability of 0.6 whatever, it's not very high. But if we are willing to expand the interval within which we expect our random variable to fall, then the probability of this obviously is increasing. So in many practical cases, this probability of 2 sigma is basically taken as a standard, so to speak. So whenever somebody is saying that, okay, this particular machine, let's say, which is manufacturing this particular part for something, and this part has certain dimensions which this machine is supposed to basically manufacture with. Well, dimensions might not be exactly like we have set this machine to do. It may be a little bit more, a little bit less, but we can evaluate this more or less. It depends on certain circumstances. Maybe it depends on different voltage in the electrical supply. Maybe it depends on some kind of shaking the table somewhere where this particular machine is installed. I mean there are different random factors which really affect the precision and the measurements of the part which we are manufacturing. So the people who are making the specification for this machine, they can say that in 95% of the cases the dimensions of this particular part will be from 2. And if this from 2 is not very large, then we can say, okay, I like this specification. The 2 sigma is pretty good enough. And if there are certain details which are outside of this range, which means their dimension is really wrong, well, we can throw them out, but considering this probability, I can say that I will throw out no more than 5% of my production as just no good or I will just sell it cheaper or something like this. But the quality ones which I define as this particular interval, the quality will be here. Now some other manufacturer can come in and say, you know what, the normal distribution of the dimensions of my detail, of my part which I am manufacturing, they also adhere to a normal distribution, but with a significantly smaller sigma, significantly smaller standard deviation. So the graph of my parts would be this. So more concentration will be around mean. The mean is supposed to be the gold standard, this particular part is supposed to be, right? So with a greater sigma, we can actually say that with this one, we can have this interval which is sufficient for us because the second machine is smaller than the sigma for the first machine. So we can get a wider interval. So the second machine with a probability of 99% gives me the good result, the part which has the good dimension. So something which is supposed to be put aside as a bad quality product would be in only less than 1%. So if I have to choose among these two machines, I would choose the machine which gives me my good dimensions with the higher probability, right? So that's basically how we are using these formulas just to evaluate the range and the probability of the random variable to fall within this range. If you like it, then we are buying this particular machine. If we don't, we are looking for something else. All right, so these are so-called sigma limits, one sigma limit, two sigma limit, and three sigma limits. So for normal random variables, these are very, very important. So we have to know the necessary validity of our prediction and use the sigma to basically achieve this particular level of our prediction. If we require that our prediction should be very, very precise, which means we have to really have a higher probability of this prediction to be true, then we have to really choose the three sigma interval. And if that interval is good enough for our practical purposes, then we accept that particular manufacturer and facilities which do whatever we want with this particular precision, with this particular probability. So that's it for today. I do suggest you to read the notes for this lecture on Unisor.com. That's accompanying every lecture. And basically that's it. Thanks very much and good luck.