 So, what is it? Next one is the 28th. Gosh, it's almost May. And, of course, there's going to be one at one other homework that's going to be a week from that, and that's probably going to be our last class. Okay, so... Yes? Is it going to be circulated from one? From example to take home. So, there won't be any issues with that? Yeah. So, what I handed in here is handed out is partial solutions to homework. So, I kind of give you the solution to part A of each problem, I hope. Thank you. Well, so, you still have to fill in a few blanks. I mean, it's one of those classes where you have fill in the blanks, right? Those are nice classes. So, I left like an numerical computation. So, we'll talk about this a little bit today and just give you a head start. And, we'll continue to talk about probability models. Certainly, we need to talk about continuous probability models. So, continuous random variables. And then, look at some of these applications. Some of the concrete applications. But let me say from the beginning, some of you have started looking at the homework, is that there's, again, a learning curve as to how to interpret these problems and how to, you know, successfully solve them, right? You have to develop sort of the model. So, it goes back to pretty much that five-step method. You start by reading the problem, digest it, then you start building the mathematical model, right? So, you decide what tools you need, right? Well, we're in the chapter where probability models. So, it is either discrete or continuous models. Then you build it, right? You build the model and you build the model by selecting variables, okay? Selecting parameters, figuring out what are the assumptions, what is known, what is assumed, right? What is something that's certainly given to you, setting stone, what's something that you have to infer, you know, make assumptions. Then use one of these kind of heavy machinery that probability theory gives you. So, we're going to talk about central limit theorem today, right? Solve the model, interpret the results, come up with your kind of lay, you know, explain the conclusions. And I think it will depend on how much exposure to probability you've had and being able to accomplish each of these steps successfully, okay? So, if you didn't have too much probability, then it might be a little bit difficult to, you know, argue that something is, you know, there's only one correct answer versus there could be, you know, possible different interpretations for the same question or for the same problem, okay? And as you can see, I mean, as you can imagine in deterministic systems until now there was little room for, well, for interpretation. I mean, there was, if you knew you had to come up with a deterministic model, then, you know, that was pretty much it. And then you, so we didn't have too much to debate on. But here it's more of the nature of the problems where there is uncertainty, right? There is randomness that you're going to get certain things that may or may not be intuitive to you. And so, if they're not intuitive, you're not going to be able to give a very good interpretation to the outcomes, okay? So, I think the way to approach this process here is to try to see how these kind of problems get, how these kind of, how these models get built and sort of make a epsilon step and try to apply these skills to maybe similar situations, okay? But unless you know a lot of, you know, stochastic modeling for which we have a different course, right? Then, so know the limitations. We're not going to be able to move too far away from the type of examples that we talk about, okay? So, I hope that's useful. Well, this comment should be helping you in kind of dealing with the problems we talk about. So, first, let's set the stage here. So, we're going to talk about continuous random variables. And the random variable still remains to be a function. So, we're still going to talk about, sorry, I'm switching from s being the sample space to omega. And there's a reason for that, but it's not essential. But it is to indicate that now the sample space is likely going to be, you know, the number of outcomes is going to be not just the discrete sets, the set of outcomes. So, space of possible outcomes of a certain experiment, right? So, it may be actually something in the plane. It may be something on a line. It may be something, right? Not only that, but you have a function that takes every, right? So, every element in this, every outcome gets assigned a certain real number. So, you make an observation for each outcome. For each run of your experiment, you make a certain observation. That's a random variable. Now, in the continuous case, first of all, what do we, what will mean that this is a continuous versus a discrete? So, x's needs to be a nice function. And by nice, we kind of mean it's, there's a more technical term called measurable function. But, I don't know, think about it as some sort of, it's a function that's compatible with the probability that's, that's assigned to events here. So, if I have some sort of, right, I'm going to have events in this experiment, right? This is subsets of the set of all possible outcomes, right? So, if I have an event like that, what we want is we want that, so by saying that x is nice, we say that x, excuse me, that omega, the set of omegas for which x of omega belongs to some interval, let's say a to b, has a probability assigned to it. So, in other words, we're going to be able to talk about the probability of this event, right? So, let's call this event a. So, you can rewrite this as follows. Kind of the shortest possible way is, sorry, the next short possible is this is between b and a, right? So, you can keep in mind where, when we talk about the probability of an event, we just simply say, we describe that event, right? So, this event would be that the random variable takes values between two fixed numbers, correct? Or the shortest way, as I said possible, in which you might be able to write this is this. So, it's like you don't write the little omega anymore, right? No, this is for continuous. Of course, it's also true for discrete, but for discrete, this is the fact that I'm looking at the... So, the fact that I'm looking at this, so I'm taking a set of all possible events and I'm mapping it by this x into the real line, right? That's what a random variable is, right? And then I'm saying, maybe this... I'm interested in when this random variable takes values in this interval, right? So, I'm looking at all possible outcomes here that actually are mapped in that, right? So, that's how I make this. This would be the event omegas such that x of omega belongs to AB, right? By the way, another way of writing this is the inverse, x inverse of the interval AB, okay? Now, if the random variable were discrete, then in this interval A to B, x would only take accountable, let's say, a finite number of values, right? So, then you could reduce this event to a union of the events when each of those values is actually being achieved. So, again, it makes sense to think of this preimage is called of this interval in the sample space, especially when the random variable is continuous. That is, the range of values of x is an interval or something, okay? So, what would be an example like that? Just think that you have a pen or some dot, something really small, right? You've got a pen, right? And you just throw it on the table, right? That's an experiment that's random, right? You're never going to get to a specific point. And then you can say, well, I'm going to take the random variable to be what? Give me an example of a continuous random variable. For instance, it could be the location of where this pen lands, right? As measured from that edge or something. So, then it could actually have all possible distances between whatever, 0 and 10 centimeters, right? So, it's a continuous range. If you counted how many times the pen bounced. No, the first contact or something. The simplest possible event, right? The simplest possible experiment, yeah. Well, you're starting, you have to start somewhere. Maybe it's, yeah, maybe it would be a distance that it bounces back or something, you know? So, it could be some numerical value that you assign to each run of your experiment, right? So, that makes it a continuous range. Or it could be just, well, what I just said is actually you can, I only said it's, excuse me, I said that you can actually throw it anywhere on this line, on this plane, but think about it on a line, right? You throw it on a line where it hits, you know, that's a random variable, right? And then you can talk about the probability that that random variable is between certain intervals, right? A certain range. So, for instance, if you have an interval that's very, very small, the probability is going to be small that you're going to hit that interval, right? Whereas if you have an interval that's large, it's going to be a larger probability, right? Still less than one because, as I said, you have to assign a probability to that event, to each such event, right? Okay. And there are some things that I won't be able to talk about this, of why this is, you know, how come not every function like this is nice? Well, the reason is, oh, that's kind of hidden, but it says that not every subset of that sample space may have a probability assigned to it. So you, right? When we talk about discrete sets, if previously we talked about a set, a sample space which was, right, let's say a finite or a countable many or something, then you know what all the subsets of that set is, right? So you know all the events and you can, if you do counting, you know how to assign probability. Whereas if you have a continuum set of outcomes in your experiment, then not every single subset of that continuum may be, may have a probability assigned to it, right? So your experiment should include also the, so the setups should include the probability, right? So every time, so we need a sample space and a probability that assigns probability measure for a set of events, okay? Anyway, so this kind of, this is kind of the starting point in this modeling. You need to identify these things. And of course you need to identify the random variables as well, okay? Okay, so let's see what do you do or how do you describe random variables? So maybe I'll shade this so you can understand this mapping here. So in a way, the analog of the counting of the, so you say an event has probability, what? The number of possible, of favorable outcomes divided by the number of all possible outcomes, right? What would be the equivalent in, for continuous, for continuous random variable, excuse me, for sample spaces that are like a real line and I'm having that experiment of throwing a dart, yeah? So what would be the probability of an interval, for instance, that you land in that interval? Could be the length of that interval compared to the length of the whole region that you, you know, so, right? So the smaller the length, the smaller the probability, the higher the length, right? So that's just an example. But you have to define that, okay? So again, the properties are the same as for the discrete. So the first one is the probability of any event has to be positive for any event A for which you assign a probability. The probability of the entire sample space is one, so that's the certain event, right? You're always, and the probability of a union is the sum of probabilities for any mutually disjoint sequence of events. Or mutually incompatible if you want. So that says that in my, you know, sample space, I need to have things that are disjoint, right? Mutually disjoint, okay? So it's not, it's not any more counting thing, but it's, if you think about this length of interval and then union of intervals, right? If I have an interval here, an interval there, an interval there, and I say what's the probability of landing in the union of, in one of the three, right? It's going to be the sum of the lengths or the sum of the probabilities. And this actually can go with infinite sequence, okay? So do you check this? No, it's oftentimes you just work with the probability space or space, a sample space with a probability assigned to the events that are, you know, that you have experience with. But these are the properties that define a probability space, basically, okay? So let's define a probability distribution of a continuous random variable. So by definition, that's a function, so the random variable X capital X. So the probability distribution is going to be a function of defined on little x for real number x, right? And it's the probability of the event that capital X is at most little x. So again, this would, the long way would be what? The probability that of the set of outcomes for which this capital X has values at most little x, okay? So this can be defined for any real x. Why? You see, even if our random variable only takes positive values, right? Let's say my random variable is a distance, right? I can still define the probability distribution for negative little x's, right? What will be that value? If I have a random variable that's always positive. I'm looking at the probability of the event that x is less than negative 2. That's an empty set. That's an empty event, right? So it's an impossible event. So each probability is 0, right? So this function is just simply going to be 0 from negative infinity all the way to wherever the first value of the random variable takes place, right? So this again is a function from R to R. Actually, it's true that it's from R to R, but it's, you can see that being a probability, it's actually the values of the probabilities are between 0 and 1, yeah? So it's a function, it's a function, and it has lots of properties, like I think it's right continuous or something. So it is right continuous. It's not necessarily continuous. Well, maybe I'll give an example, but either right or left continuous. I think it's right continuous. So the best way to understand this is to graph it, right? I mean, if you can graph it, that's what you do to understand the probability distribution. It is similar to, well, it's not quite similar to the histogram in the discrete random variables when we said, when we said, I'm going to plot the number of other probability that the event takes a certain value, right? One, two, three, four, like, right, discrete set of values. And then I have this graph. Here, there is something similar to that, but because it's continuous random variable, then you have a continuous range, right? So here's an example. So there are some very important properties of f. So one, we said this, so we said that this is always between zero and one because it's simply a probability. One, the more important one is f is increasing function. Okay, so what does it say increasing? It says that if I have two values for little x1 and little x2, you can compare the two events. The event when capital X, our random variable is less than x1 and the event when the random variable is less than x2. One is containing the other as sets of outcomes. Which one is where? Which one is containing which? If one of them happens, the other one must happen as well. It's just a matter of saying it back in your head is in words, basically. So you're saying this is including this? This is including this, right? Because if I throw the die, right? If I run the experiment once and I measure the random variable, right? And it's less than x1 because x1 is less than x2, it's also going to be less than x2, right? So every outcome in this set, in this event, is also going to be an outcome in this event. Now, because of this inequality, inclusion, and of course the probabilities, the properties of the probabilities. If I have a smaller set and a larger set, both of them are events, in two events. Then the probability of the smaller set is less than the probability of the larger set. So this just says f of x1 is less than f of x2. Of course, for x1 less than x2. So this says that f is increasing. So it's always going to have this pattern. Of course, as I said, it might actually be zero for a long time and increase and possibly be one after one, right? For instance, if the actual random variable has values in some interval, bounded interval, right? And the third property is f is continuous, f is right continuous. What does it mean, right continuous? The value of x at a certain x is the same as the value at the limit from the right, which again is the limit as y approaches x, y greater than x, right? So the picture here is that something like this. Of course, one is here, zero is here. But at some point it doesn't necessarily have to be, it can have jumps. Of course, only increasing, it can only go up with jumps because f is increasing. But I think the value at that end point is going to equal to the limit from the right. And again, this follows from that property that you can take in property three here. That you can take unions, countable sets. Actually, it's worth to put up another property, which is that not only that f is between zero and one, but also the limit at infinity of f is one and the limit at negative infinity of f is zero. So it really says that it goes, it doesn't increase and then stay below one. It goes towards one. Again, that's, again, a property of that union, probability of the unions is some of the probabilities, right? Am I saying something totally new here or have you seen this in previous places? I mean, we could spend time and go through this, prove all of these things, but I think it's not one of our goals here. Okay, so let me just introduce one more thing that's probability density function. So that's to compare, to contrast with the probability distribution. So again, of a continuous random variable, x is simply the derivative of this capital F assuming the derivative exists, right? So assuming f is differentiable at x. Now, so f being the probability that our random variable is little x. Now, keep in mind, this is all attached to a random variable. So you have a random variable, it comes with a probability distribution and it comes with a probability density function assuming that you can take the derivative. So for instance, if you have a jump, you cannot take the derivative, right? So if your probability density function distribution has a jump like probably before, you won't have a derivative there, right? But if for instance it's continuous, right? If it's continuous and increasing, there are results that say that you have derivatives almost everywhere, okay? In fact, if it's an increasing function, it always has derivatives almost everywhere, okay? So two key properties of probability density function, which are sometimes called PDFs, right, of PDFs. Well, first of all, it's a positive function if it exists. Of course, we're talking about cases when we have this differentiability. So if it's always positive, right, because it's the derivative of an increasing function. And what's more important though is the way in which you use this and use this density function to compute probability of the event, of that key event, of the event when x is between two fixed values and that's simply the integral. Now, why is that? So that's very important. Why is this? Well, what's the integral from a to b of f of x dx? Simply the integral of capital F prime, right? So that's f of b minus f of a, fundamental theorem of calculus. Yep. Okay, so that's f of b minus f of a. So what is f of b? Probably the capital X is less than b, less than or equal to b, minus the probability that r dx is less than or equal to a, right? Now, this one is a subset of this, right? It's a sub-event. I don't think we use sub-events, but this is a subset of outcomes of this. So the difference is the same as the probability of the event when it's kind of different between these two events, right? So it's when x is less than or equal to b, but not less than or less or equal to a. So it's greater than a and less than or equal to b, okay? How many of you have not seen this before? Okay, everybody saw, okay. So maybe we should just cancel the class, huh? I think I've seen it, but I don't remember. So it's a good thing to, I mean, to fix this in your memory because, you know, it's used throughout. And of course the other way to remember this is I have a function that's positive, right? So when I'm plotting little f, which is simply derivative of that probability distribution function, right? So that was, that probability distribution function was kind of increasing, so the derivative is positive. But also there's this other property that being increasing and then leveling off at one and starting from zero, it means that derivative has to go back down to zero. So, I mean, it's not always true. You could have some wiggles here, but in general that's kind of the shape of a probability density function. And then this statement simply says integrate this function to complete the area on the curve, right? The function is positive, so the area under the curve is always the integral. So that's, so the area under the curve, under this curve is exactly under the curve y equals f of x. A between, x between A and B is equals the probability of this event, okay? And also I'm going to say this, but it's not completely correct. But oftentimes, I mean, it has to be kind of a really weird distribution or density function for this not to be true. But you know, the limit, if your probability distribution function is an increasing function and kind of not with too many wiggles, then the limit of the derivative at plus and minus infinity are both zero. So it is localized, okay? In that sense. Also, what's the integral from negative infinity to infinity? So the improper integral of this density function. It's always one because this is the probability of x being negative infinity plus infinity, and that's the certain event, right? So all of this is kind of good to keep in mind now. Just a footnote, if x is a discrete random variable, so by that we mean that it take only discrete, let's say, a countable number set of values, so k1, k2, kn. So k1, if all my guys in a certain event, right? So it's the event that x takes the value k1, k2, if it's another event, right? That's another event. These are all mutually disjoint events, right? And let's say k1 is less than k2, less than k3. Then what's the probability distribution? I mean, we should say, to be very precise, probability distribution function. Okay, so it's because the function on the real axis, it's on the real line. So this would be, let's say k1 is here, k2 is here, k3 is here, k4 and so forth. So I'm plotting something that takes a discrete set of values and they're all positive. So how is the probability distribution function now look like for this one? Well, it's the probability that your capital X is less than zero. So zero and zero, zero, zero all the way to k1, right? Now what's the probability that the random variable is less than or equal to k1? Well, it's going to be exactly the probability that X is k1, right? And that's the probability of the first event. Exactly. So it's going to be a positive, well, right? It has to be positive otherwise you wouldn't list it. So it has a jump there, right? And then it stays constant up to the next one, right? What's the probability that X is less than or equal to k2? Well, it's really the probability that X is k1 or X is k2, right? Which it really means is the probability of the union. And this being disjoint events, probability of the union of two disjoint events is the sum of the probabilities, right? So it's another step, right? So this would be the probability of a1 and the probability of a2. So every time you have a step by that amount, right? Now if you have an infinite set of values that the random variable can take, then this is going to continue and eventually kind of converge to one, right? If it's a finite set of values that the random variable can take, it's going to find a number of steps and then just going to level off to one, okay? So for instance, for this kind of probability, for this kind of random values, it doesn't quite make sense to talk about the density function, right? In density function we mean the derivative of this piecewise constant function. So it would be zero everywhere except at this discrete set of points, right? So the density function is important for the case when you don't have this kind of jump. You don't have a discrete set of values, but the rather continuous set of values. Okay, so let's really quickly talk about expectation of a continuous random variable. We talked about the expectation of a discrete random variable, but now I have something that takes a continuous range of values, right? And here's a case which is kind of the other extreme. It's not discrete, so it has a density function with a probability density or PDF function, which is little f of x, right? Then the expected value of this random variable is defined to be, again, it's this symbol which says summing up all the values of x, multiply with the respective probabilities. So it's an integral, but in this case it can be written explicitly in terms of the probability density. So it's integral of little x f of x dx. The way to think about this is almost like a change of variable. Let me go through two steps here. So this would be the integral of px of omega dp of omega. Let me leave dp here. Now, if I'm going back to the real axis, I'm going to replace this with little x. And this would be kind of hard to explain this without going through the integration process explicitly. But this is, I don't know, that's not going to be helpful. Maybe skip all these steps until you see the formula that one uses most of the time. So the expectation of x is x little f of x dx. Again, I cannot justify very well all these change of variables that took place here because we haven't talked about measure theory much. So maybe this is digestible. It's basically an integral of x with respect to f of x. But if not, then this is just little f of x or f prime, capital f prime dx, right? So this is just going to be... So in essence, the way you complete the expectation is you multiply x with the probability density function and you integrate between its negative infinity to positive infinity. And this is consistent with the definition for discrete random variable. What was the definition for discrete random variable? It was, right, let's say it takes the values k1, k through kn. There will be kn times the probability that x takes the value kn, right? So this would be like pn. So this is the summation of kn, pn. So instead of the summation you have an integral, and oftentimes you can actually take a limit, a continuous random variable is the limit of a sequence of discrete random variables. So those kind of are well defined. And that's the variance of a continuous random variable with probability distribution f. Well, it's v of x of... v ar of x is the integral from negative infinity to infinity of little x minus m squared f of x dx. Where little m is actually the expectation of capital x. So this is, again, this is just the expectation of the x minus m squared. And once you say, what's the probability density function of this random variable? x minus m squared is the same as the probability density function of x. So it's this f, right? So if you start from here, you... Well, I don't know. Again, there are some kind of a jump from one to the other, right? Capital x gets replaced by little x. This is the real number. Little f of x shows up here in an integral from negative infinity to infinity, right? Is this novelty to any of you? It's okay if it is, but... So if you have a random variable and you know it's probability density function, then this integral is going to give you the variance, okay? Again, this matches the case of discrete random variables, which would be... It's a serious, right? So oftentimes this looks like a lot better computation than a serious computation, right? Okay, so actually there's even more. So there's an even more general formula, which if you have a random variable x and you compose it, you apply another function to it, right? It's a composition. So the one above would be... Or think about just taking the square of capital x, right? There would be a new random variable. And you take the expectation of that. Well, anybody guess what this might be in terms of the integral density function? The density function. I'm not saying what g is, just... But if you want to think of g as x squared, then that's fine. If g is the square function, then this would be actually the square x squared. Okay? Little x squared. If g is sine function, this would be sine of x, okay? So it really comes from that change of variables, but... So in general it's just going to be g of x, g of little x times capital x, okay? So now you can match this maybe even better with that, right? So the expectation... So here the function g would be little x minus m squared, right? So that's what gets multiplied by f of x. Okay, so... Of course this is used in... For what's known momentum. So moment, excuse me, of random variable. You could talk about the square of a random variable, right? And talk about the expectation of that. That would simply be integral of x squared f of x, right? And so forth. So I was joking, sine of x, I don't know if you'll ever see, but powers of, right? A random variable that is sometimes important to compute, so... Okay? Certainly the second power is important. Further powers are not so important. Okay, so let me say one more thing and then list the central limit theorem, okay? So a very important example of a density function is the Gaussian density function, or which implies a normal distribution function. So it's just like that Poisson distribution that we said, here's a distribution, okay? We didn't say what the experiment was. We didn't say what the random variable was, right? We just said there are important random variables in generic experiments for which that was a distribution, right? So same here. Probably it's good to think about, if you haven't seen this before, is that there are certain, well, there are actually universal experiments and corresponding random variables which share this kind of special density function, and that is we're going to use this notation. So there's a sigma here, so sigma in denominator. So just to familiarize yourself, if I take... excuse me, m equals 0 and sigma equals 1, that would be kind of the normal distribution. Then g of 0, 1 of x is going to be what? 1 over the square of 2 pi e to the minus x square over 2. Yep. Exactly. So sigma will end up being the standard deviation for this random variable, for this distribution. Again, just to see the shape, this is the special case, right? That's the normal distribution. And the one there is just a translate and possibly a change in scale in the x-direction. So typically this is going to look... of course, this is g 0, 1. So this is the standard... this is the normal distribution. And it's an even function. It has a peak at 1. And what's the value? It's 1 over 2 square root of pi. What's the property? The important property that one has to check is that the integral of this actually ends up being 1, right? So that's basically where this factor comes from, because when you do the computation of just the integral of e to the minus x square over 2, integral of that over the whole real line, you end up with a square root of 2 pi. So that's an exercise you should have seen in Calculus III. It's a nice... it's a change of variable to polar coordinates. So look in a Calc III book where it talks about a change of coordinates from rectangular to polar coordinates in the plane, and this is going to be there, as an example, as a computation. And it's a very good exercise to do. Okay, so again, that's the reason why they squared up 2 pi, right? It's because you want a probability density function to integrate to 1. Okay, so... Yeah, if I have time, I'll show you, but maybe it's, you know, you should go and try to look for that computation on your own. Now, of course, what will be... what will be for general m and sigma, the thing is just going to be shifted it's going to be here and have the same shape, of course, but it's going to be shifted and it's going to have... it's still going to have the integral to be 1. Yeah? I mean, it has to be. So that's where... that's the reason for that factor in front, right? In the general. And the factor was 1 over sigma times 2 squared up pi. So this will be the 1 over sigma squared of 2 pi. The peak of this thing is not so important. The more important thing is actually the inflection points here. So if you're... if you're really into calculus 1 stuff, you could take that function differentiated twice and you're going to see that the inflection points, there are two inflection points and they happen at plus or minus 1. And for this thing, they're going to happen at sigma units away from the mean. Okay? And let's see. So if that's the case, can you see the expectation of this? So two things, two computations that we're not going to... I'm going to just indicate but not actually complete them. The expectation of such a random variable is going to be the integral of x times that, right? And remember what I said, it has to do with that center of mass. So if I have that thing, what's the center of mass of this... of the area under the curve? If the curve is like the symmetric like this, it's going to be in the center, right? And where is the center? The center is at m, right? So that's... So this, again, this is a substitution that you do in the integral and you end up with... So this is by substitution rule, right? If you don't do so on your own, you're not going to appreciate the statements. As well as the variance, which is, you know, the variance is going to be little x minus m square g of m sigma. And again, this is going to take some... not a lot of work, but it's going to take some integration by parts and you will end up... you will see that this is actually this exactly sigma squared. Okay? So this means that what's the standard deviation of x is precisely sigma. Okay? And of course, the mean is precisely m. So that's... Okay? And let's just say one last thing is that... So let me just say the only thing is that is that there are two properties of this standard deviation. One is that the area under the curve between... within one standard deviation from the mean. So it's simply a computation and it's almost 68% or it's close to 68%. So this is one standard deviation from the mean. That's saying in terms of... What is it saying in terms of probabilities? It simply says the probability of a random variable with this density function of this random variable being between those two values is 68%, right? Or close to it. So it's not an exact... Yeah? Okay? So it's always important in this to be able to translate back to probabilities. Also, two standard deviations from the mean. That's 95%. Three standard deviations, probably 98% or something. 99.7%, thank you. So this is two standard deviations. And again, this means the probability of the random variable between m minus 2 sigma n plus 2 sigma is 95%. And you can keep going on our own, right? Well, 95% is kind of an important number here. It says that this is 95% confidence interval. So we're going to have to finish here for now, but this is the place where we can start talking about the central limit theorem, which is kind of dwarfs the strong law of large numbers that we talked about previously. Because it gives additional information into what happens with a sequence of random variables that are identically distributed and independent. So unfortunately, we're going to quit there for now, but if you would like to start taking a look at these solutions and the problems that I assigned and I think you should be able to do number well, you can look at number 1, as the first problem exists too, and involve a central limit theorem. So I would say try to read the central limit theorem from any probability book or no, it's not online, but I'll try to put it online. And this should help you kind of get started at least interpreting how the problems should be put into equations and stuff. And I think yeah, normal problems involve central limit theorem. Okay, thank you.