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Transcript
00:00:390Paolo Guiotto: Okay, good morning. Sorry for the little delay.
00:08:490Paolo Guiotto: So, I realize that we have also
00:11:750Paolo Guiotto: One day that we won't have class
00:15:140Paolo Guiotto: in December, because of the medicine test.
00:19:280Paolo Guiotto: So… We have one lecture less.
00:24:620Paolo Guiotto: This means that I will, I will cut that part. I was, I started last time on the space of sequence. If you are curious, you can read. It is not a topic, for the exam, but…
00:43:210Paolo Guiotto: It was interesting to show you how to build a probability space
00:49:30Paolo Guiotto: From, an infinitely dimensional probability space,
00:55:960Paolo Guiotto: Which is a sort of prototype of the Brownian motion. So, let's say that we know that the probability space is just a measure of space.
01:09:460Paolo Guiotto: Where the total probability is 1. And so this means that all probabilities of a measure are verified for a probability measure. In addition, we have the continuity from above, which is always verified.
01:28:230Paolo Guiotto: Okay, now, today we start moving into the… some of the topics, more typical topics, tools of probability, and in particular, we start talking about random variables.
01:51:640Paolo Guiotto: Well, a random variable is just a measurable function. So, definition… Omega… F… B… Be a probability.
02:09:979Paolo Guiotto: space, huh?
02:12:280Paolo Guiotto: a function a measurable function.
02:23:270Paolo Guiotto: Here, random variables are usually denoted by uppercase letters, like X, Y, uppercase X.
02:32:280Paolo Guiotto: capital X, capital Y, etc. So, it's a function X defined on omega, real-valued.
02:40:700Paolo Guiotto: So a measurable function X is called… random…
02:52:370Paolo Guiotto: valuable.
02:54:980Paolo Guiotto: So, it is not a new, any concept, it's just the concept of measurable function.
03:02:620Paolo Guiotto: With a different name.
03:06:610Paolo Guiotto: So, there are names, that are used in probability. We say that, we say…
03:21:930Paolo Guiotto: That's… For example, X is positive, Instead of, fate.
03:34:950Paolo Guiotto: Of, say, almost everywhere, we will see almost Surely.
03:43:630Paolo Guiotto: Il faux.
03:47:340Paolo Guiotto: Well, it should be, X greater or equal than zero almost everywhere, that is the probability.
03:54:180Paolo Guiotto: that X is non-positive, less strictly negative, is 0.
04:02:660Paolo Guiotto: Or in this case, equivalently, since the total probability is 1, we could also say if the set of omegas where X of omega is greater or equal than zero, this has probability 1, so it is a sure event.
04:20:720Paolo Guiotto: So this is,
04:23:800Paolo Guiotto: The shortening of this is A.S.
04:28:310Paolo Guiotto: And this is the same of almost everywhere.
04:31:630Paolo Guiotto: Now, for, X,
04:40:00Paolo Guiotto: for X in a random variable X in L1 omega, it is well-defined.
04:54:810Paolo Guiotto: The expectation, or the expected value, E.
05:00:300Paolo Guiotto: expecting,
05:06:400Paolo Guiotto: of X, or expected value.
05:17:550Paolo Guiotto: of X, or mean value of X, these are all synonymous, which is defined with this symbol, E of X.
05:27:640Paolo Guiotto: And it is, the integral on AXA.
05:32:430Paolo Guiotto: of X on the set omega with respect to the probability p. So it is the integral of X.
05:42:980Paolo Guiotto: Also, another important quantity, that is defined for X, actually in L2, for…
05:53:530Paolo Guiotto: This requires that X be in L2, Sierra, we…
06:00:40Paolo Guiotto: I just return on this in a second, that we should notice that if axis in L2
06:06:20Paolo Guiotto: In general, it is not in L1, but in the case of probability spaces, these spaces are included, so L2 is contained in L1, and so it is, in particular contained in L1, so the expected value.
06:21:950Paolo Guiotto: Makes sense for this. But for this, makes it… makes sense also what is called the we call…
06:32:110Paolo Guiotto: variants.
06:37:750Paolo Guiotto: of X, the quantity, let's say, V, of X.
06:44:660Paolo Guiotto: defined formally as the expected value of X minus its expectation.
06:54:650Paolo Guiotto: squared.
06:56:240Paolo Guiotto: So, it is the average of the… Say, of the…
07:03:330Paolo Guiotto: displacement between X and its average.
07:07:780Paolo Guiotto: So it's a way to, to measure how X is non… is, far… is… is spread respect to the expected value.
07:20:330Paolo Guiotto: Now, I noticed this, that if it is in L2, it is also in L1, because in this formula, you have the expected value of X, which is defined when X is in L1.
07:33:400Paolo Guiotto: You can easily check by doing a simple calculation, developing the square, that this is the same of expected value of X squared minus the square of the expected value of X.
07:49:800Paolo Guiotto: sometimes it might be useful to know this. So this is the remark that says that L2 is contained into L1,
08:04:870Paolo Guiotto: Or… a, probability.
08:10:760Paolo Guiotto: Mobility measure.
08:14:190Paolo Guiotto: And moreover, which is not important at this point, but it will become important later. The L2 norm.
08:23:710Paolo Guiotto: The L1 norm is controlled by the L2 norma, so in other words, the L2 norma is stronger at the end than the L1 norm.
08:34:200Paolo Guiotto: for every X in L1.
08:37:20Paolo Guiotto: This is a simple consequence of the Cauchy-Swartz inequality.
08:42:350Paolo Guiotto: these… Follows.
08:47:910Paolo Guiotto: from… the Cauchy Schwartz inequality.
08:53:930Paolo Guiotto: Because, the, if you want the one norm of X,
09:00:680Paolo Guiotto: is the integral on omega of the modulus of X.
09:07:920Paolo Guiotto: DP.
09:10:660Paolo Guiotto: And, now you can always think this as a product between 1 and the modulus of X, and then you apply the Kuchish-Wall's inequality. It says that this is less than integral omega of 1 square dp.
09:24:900Paolo Guiotto: To one half.
09:27:150Paolo Guiotto: times the integral on omega of modulus X squared, so X squared, DP?
09:34:850Paolo Guiotto: to 1 half.
09:36:560Paolo Guiotto: Now, the first quantity, since you are integrating 1, it's the measure of the set, so it is the probability of omega, which is by definition equal to 1, so to exponent 1 half, it is equal to 1.
09:52:400Paolo Guiotto: And the other one is the L2 norm of X.
09:57:580Paolo Guiotto: And here we have the inequality, so this is less or equal than 1 times DLC norm of X.
10:07:20Paolo Guiotto: as, claimed.
10:09:130Paolo Guiotto: So this, in particular says that if you are in L2,
10:12:910Paolo Guiotto: you are also in L1, and so, as we say here, if the variable is in L2, the expected value is well-defined, and the variance makes sense.
10:27:990Paolo Guiotto: Another quantity that, that,
10:33:310Paolo Guiotto: It's, important in probability is the so-called covariance of two random variables. So, if,
10:43:40Paolo Guiotto: X and Y are… L2.
10:47:760Paolo Guiotto: Omega, random variables, D, there, covariance.
11:01:420Paolo Guiotto: is the quantity.
11:04:460Paolo Guiotto: is, by definition, say, C.
11:08:830Paolo Guiotto: XY.
11:11:510Paolo Guiotto: Equal.
11:13:130Paolo Guiotto: We can say expected value of,
11:17:830Paolo Guiotto: X minus its expected value, its average.
11:22:760Paolo Guiotto: its VIN value times Y minus its expected value.
11:31:430Paolo Guiotto: Since the two are in L2, the product is in L1, so everything here is well-defined, again, and you can easily see from the cohesive parts inequality, so by
11:45:680Paolo Guiotto: Because she thwarts inequality.
11:49:510Paolo Guiotto: We have this bound that the absolute value of the covariance
11:55:730Paolo Guiotto: Well, maybe it's better if we write a COV.
12:01:630Paolo Guiotto: So, the models of the covariance Between X and Y.
12:09:380Paolo Guiotto: If you put the modulus on that expectation, it is the expectation of a product.
12:17:690Paolo Guiotto: And, because of the Cauchy force inequality, This is, applied to these two.
12:26:820Paolo Guiotto: variables. This is less than, D, if you want the integral.
12:33:800Paolo Guiotto: Well, we have to switch to the probabilistic notation, where you never see an integral, but you always see expectations. However, if you want, we can see, because Schwartz inequality is modulus of integral omega of, let's say, Z times WDP,
12:53:930Paolo Guiotto: This is the quotes written in the language of integras. This is less or equal integral of modal z squared.
13:03:110Paolo Guiotto: To explain one path.
13:06:180Paolo Guiotto: times integral omega of modulus W squared.
13:10:870Paolo Guiotto: exponent 1 half. But all this can be seen also in the language of probability, in the… in the, say, in the
13:20:890Paolo Guiotto: In the grammar of probability, this is modulus of the expected value of Z times W is less or equal than expected value of modulus Z squared to exponent 1 half
13:36:180Paolo Guiotto: times expected value of modulus W squared to exponent 1 half. So, this is…
13:43:640Paolo Guiotto: how the Cauchy-Swartz inequality looks like with the probabilistic notations. So, we can suppress the integral here and say just it is less or equal than the expected value
13:58:260Paolo Guiotto: of the modulus of the square there, so the square, X minus EX. Here we consider mainly real-valued stuff, so the modulus of the square.
14:10:310Paolo Guiotto: The module squared, or the square, are the same thing, to exponent 1 half
14:17:610Paolo Guiotto: times the expected value of Y minus Expected value of Y.
14:26:10Paolo Guiotto: Squared to 1 half.
14:29:590Paolo Guiotto: And these are exactly what we defined as the variance. This is the variance of X,
14:36:490Paolo Guiotto: 2 exponent 1 half.
14:40:280Paolo Guiotto: And this is the variance of Y.
14:43:750Paolo Guiotto: to exponenta.
14:45:900Paolo Guiotto: one class.
14:47:380Paolo Guiotto: So we have these models of covariance.
14:52:390Paolo Guiotto: Between X and Y.
14:55:630Paolo Guiotto: Is less or equal than the variance.
14:59:20Paolo Guiotto: Ovaxa.
15:00:620Paolo Guiotto: Excellent. One path.
15:02:710Paolo Guiotto: and variance of Y to exponent 1 half.
15:08:740Paolo Guiotto: This quantity, they have… they deserve themselves a name. The root of the variance is… it is called the standard deviation of the random variable, so this is called the standard
15:25:690Paolo Guiotto: deviation.
15:31:20Paolo Guiotto: of XM.
15:39:660Paolo Guiotto: And the other one, the standard deviation of Y.
15:44:650Paolo Guiotto: So, in particular…
15:48:860Paolo Guiotto: Suppose… assuming that these variances are different from zero, well, notice that the variance of a variable is equal to zero.
15:59:150Paolo Guiotto: The variance equal to 0 means that the expected value of this quantity, X minus expected value of X,
16:09:500Paolo Guiotto: All this squared equal to zero.
16:12:400Paolo Guiotto: Now, remember that this is an integral, so what we are saying here is integral on omega of that square in the measure P is 0.
16:23:580Paolo Guiotto: And since this quantity being a square, is a positive quantity, integral equals zero means that the argument in the integral is zero almost everywhere, with respect to P, or in the new language, we would say that X minus E of X
16:43:810Paolo Guiotto: Squared is equal to zero, almost surely.
16:50:990Paolo Guiotto: Or another way to write this, is, or equivalently, We say, with probability
17:04:619Paolo Guiotto: equal to 1. So when we say something is true with probability 1, it means that the probability of that event is 1,
17:13:200Paolo Guiotto: And it means that that event is verified almost surely. But this means that that quantity
17:22:410Paolo Guiotto: Being 0 means that this is 0, and therefore X coincides with its expected value, almost surely. So, X is equal to the expected value of X, which is a constant, that's a number.
17:38:720Paolo Guiotto: Okay, so, almost surely… And this means that X is… constant.
17:51:380Paolo Guiotto: Almost, surely.
17:56:640Paolo Guiotto: So, variance equals 0 is a particular case, is the case when the variable is a constant. If the variable is not a constant, the variance won't be equal to zero. And therefore, if we assume that X and Y are both non-constant.
18:15:50Paolo Guiotto: We can divide this inequality here by the right-hand side and get this. So…
18:24:380Paolo Guiotto: if the variances of X and the variance of Y different from zero.
18:35:150Paolo Guiotto: We have these models of the covariance between X and Y.
18:41:640Paolo Guiotto: And, they, divided by the standard deviations of X and of Y.
18:52:590Paolo Guiotto: It is a product of their roots.
18:56:340Paolo Guiotto: Of the variances. This is less or equal than 1.
19:03:100Paolo Guiotto: Now, this quantity here, Takes another name here, this quantity.
19:10:870Paolo Guiotto: is a number between minus 1 and 1, and this is called the linear correlation of X and Y. So usually it is denoted with the raw XY, or equivalently, raw XY in the sub-index. This is called the
19:30:470Paolo Guiotto: linear… correlation.
19:39:140Paolo Guiotto: Off.
19:40:660Paolo Guiotto: X.
19:41:790Paolo Guiotto: And… Why? Or… Another name is the Pearson correlation. Pearson is the…
19:50:760Paolo Guiotto: Name of a statistician who introduced this.
19:55:90Paolo Guiotto: Mmm… okay.
19:59:970Paolo Guiotto: Well, remind that at the end, this inequality comes from what?
20:07:530Paolo Guiotto: comes from the…
20:09:550Paolo Guiotto: It is this one that comes from the Koshichworth's inequality. It is the Kaushich-World's inequality, but written with this other emphasis.
20:19:630Paolo Guiotto: Okay.
20:23:130Paolo Guiotto: Now, An important… there are some important, concepts that are attached with the Any random variable.
20:36:260Paolo Guiotto: The first important is the concept of low off… random variable.
20:46:940Paolo Guiotto: We may say at the end that, a random variable, For the probabilistic purposes.
20:54:750Paolo Guiotto: is known once, you know, it's low. So, in practice, you don't need to
20:59:850Paolo Guiotto: to have a sophisticated modeling, because the law is actually a probability measure on, on R, basically. And the idea is the following, no? So, suppose that, we have, omega F…
21:18:120Paolo Guiotto: B… Is a probability space.
21:25:900Paolo Guiotto: Then we have a random variable X.
21:31:940Paolo Guiotto: So, this means that, this means that exit is,
21:40:650Paolo Guiotto: Let's use the old, terminology, is a measurable function.
21:50:20Paolo Guiotto: So this means what? That, you remind the definition of measurable function means that
21:55:470Paolo Guiotto: The set of omegas, so the…
21:59:00Paolo Guiotto: The elements are in the domain.
22:02:300Audio shared by Paolo Guiotto: of XM, so the omega.
22:06:470Paolo Guiotto: In the sample space here, we would say capital omega, such that X of omega belongs to some interval I,
22:15:170Paolo Guiotto: This set here, which is a set of omega, is… it belongs to the family of events, so… to which we can assign a probing to the family F, belongs to F for every i contained in our interval. This is the
22:33:950Paolo Guiotto: definition of measurable function. So, in particular, it means that we can compute the probability of this set. Now, instead of writing all this thing, omega and capital omega, such that X of omega belongs to I,
22:53:370Paolo Guiotto: which is a very long, but precise, of course, writing, we just use this shortening, which is the probability that X, capital X, belongs to the interval I, which is a little bit more
23:09:80Paolo Guiotto: efficient as notation, because it says you want to compute the probability that that X, that can represent whatever you want, belongs in that range. So, for example, you take I equal interval AB, this means probability that that random variable, capital X, is between A and B.
23:32:440Paolo Guiotto: What is the probability that the range of X is in between these two values, A and B, no? So, it's an interesting quantity.
23:42:550Paolo Guiotto: Okay, now, if we focus on this, so we can say that to each interval, we can assign, somehow, a number, which is the probability that X belongs to that interval.
23:59:80Paolo Guiotto: Now, this is… sounds like a set function, no, if you think about it. You have a set i, and you assign a number, which is the probability that X belongs to that i. That number is a number, since it is a probability, it is between 0 and 1,
24:14:480Paolo Guiotto: So, it has the same kind of nature of a probability function, no? A probability function is a function that assigns to every set of certain family, a number between 0, 1,
24:29:120Paolo Guiotto: In such a way that it is a measure. So probability of empty set is 0, and the probability is countably additivity, and probability of the full space is 1, no?
24:39:160Paolo Guiotto: So we may say that these are something of these characteristics, because if you call for a second this operation, mu X of i.
24:51:460Paolo Guiotto: You would say that, for example, mu X of empty is what? Is the probability that X belongs to empty set. So, what is the probability that X belongs to the set made of nothing?
25:06:230Paolo Guiotto: Now, what are the omegas for which X of omega belongs to the empty set? There are no omega, so this will be an empty event, and this is the probability of empty set, but that's a measure, so probability of empty is zero.
25:20:760Paolo Guiotto: So, mu X of empty would be 0.
25:25:610Paolo Guiotto: You would have also that mu X of the full space here. You see that these are subsets of the real line, okay? So they do not belong in some
25:37:720Paolo Guiotto: strange omega in some space. There are space of numbers, this is a set of numbers, so mu X of r means the probability that X belongs to R. But if X is a random variable, I'm saying, what is the set of omega for which X of X is a real number?
25:55:600Paolo Guiotto: every omega, no? If X is just a function, real value, so this is the probability of omega itself, which is equal to 1.
26:06:250Paolo Guiotto: Well, about the additivity, here we have a problem, no, because the family of intervals is not a sigma algebra. Even if you take two intervals and they are disjoint, you do the union, this is not an interval, no? So, we cannot pretend that this works exactly as a measure.
26:24:950Paolo Guiotto: But actually, there is a way to make this a measure, enlarging the class of set i.
26:32:440Paolo Guiotto: What is the natural class we should consider? Now, the natural class is the class
26:40:650Paolo Guiotto: should be a class that contains interval, and it is a Sigma algebra.
26:46:630Paolo Guiotto: Now, there is a unit class that does this, because that's the sigma algebra generated by intervals, if you want, and this is what we call the Borel sigma algebra.
26:58:110Paolo Guiotto: Now, the idea is that, we can… we can.
27:04:170Paolo Guiotto: Extend…
27:08:960Paolo Guiotto: MUXA, 2.
27:12:940Paolo Guiotto: set… E?
27:17:420Paolo Guiotto: That belongs to the… BR, this is the Borel.
27:26:210Paolo Guiotto: Sigma. Algebra.
27:28:930Paolo Guiotto: So, which is the sigma algebra generated by I subset in R, I interval.
27:41:980Paolo Guiotto: Soft.
27:43:40Paolo Guiotto: I take the family of all the intervals, all possible intervals. Of course, it's an infinite family. That family is not a sigma algebra, but they always know that there is a sigma algebra that contains that family.
27:58:460Paolo Guiotto: And the cheapest sigma algebra, the minimal sigma algebra, is what is called the sigma algebra generated by this family. And that's a special name of… it's called the Borel sigma algebra.
28:11:630Paolo Guiotto: Now, to check this, so, saying that, MUX OVE,
28:19:40Paolo Guiotto: will be the probability that X belongs to E.
28:24:840Paolo Guiotto: Now, to check that this works, I need to know if this set here
28:30:770Paolo Guiotto: is a set to which I can assign a probability to. Otherwise, what I've, written there is not, is not,
28:40:170Paolo Guiotto: appropriate statement. And this is, in fact, can be proved.
28:45:810Paolo Guiotto: So, saying this proposition… So let's… X… B… A random variable on… Omega F… P, the probability space.
29:05:300Paolo Guiotto: Actually, it can be proved that,
29:11:310Paolo Guiotto: This is a characterization of being a random variable. However, we are interested in this part. Then, this sets, X belongs to E,
29:23:120Paolo Guiotto: So that is…
29:25:420Paolo Guiotto: the set of omega in the sample space, capital omega, such that X of omega belongs to E. This is N.
29:37:60Paolo Guiotto: event, for every E, that belongs to the Borel sigma algebra.
29:48:470Paolo Guiotto: So, if this is, true, you see that,
29:54:550Paolo Guiotto: Once I know that that's an event, I can compute its probability, and that function mu X, is well-defined, and it turns out that it is a probability measure.
30:06:60Paolo Guiotto: In this case.
30:13:250Paolo Guiotto: MuX.
30:15:780Paolo Guiotto: is… Well… Defined… on the Borel sigma algebra.
30:26:110Paolo Guiotto: And… It is… probability.
30:34:890Paolo Guiotto: Man.
30:42:340Paolo Guiotto: Now, I will, there is the proof in notes. It's not particularly…
30:50:760Paolo Guiotto: is a typical set theory proof, how to prove that the first part of the statement, how to prove that sets X belongs to E are
31:05:10Paolo Guiotto: are events for every, set E, which is in the Borel sigma algebra.
31:12:370Paolo Guiotto: Now, this is done by steps. First, you have that this is true when E is an interval, because that's the definition of measurable function, no? If E is an interval, the set where X belongs to E is a measurable set, because this is the definition of measurable function, as we noticed here, right?
31:33:40Paolo Guiotto: So, this is true when the set E is an interval.
31:38:270Paolo Guiotto: Now, second step, one proves that the family of E, of set E, for which the event X belongs to E is in the family F,
31:55:750Paolo Guiotto: That family is a sigma algebra, so when that family is close respect to unions, countable unions, complementary, contains empty set and the full space.
32:07:700Paolo Guiotto: So, once you know that that's a sigma algebra, that contains all the intervals, it means it contains the sigma algebra generated by the intervals, and therefore that becomes true for every set E.
32:21:590Paolo Guiotto: So, let's say I don't do the proof, but the idea… of… Roof, huh?
32:33:820Paolo Guiotto: So one defines the family for which the conclusion is true. So, let G say the family of sets E
32:45:30Paolo Guiotto: Which are, in, in the Borel class, so they are Borel sets.
32:53:860Paolo Guiotto: So they are sets in this margin generated by intervals, such that the set X belongs to E is an event that belongs to F.
33:06:520Paolo Guiotto: Okay, so I take the family, for which the conclusion is true. The goal is to prove that this family is the right class.
33:17:230Paolo Guiotto: So the goal is this.
33:21:460Paolo Guiotto: G is all the morale class.
33:26:10Paolo Guiotto: Now, what can be said?
33:28:460Paolo Guiotto: The facts are… Number one.
33:33:600Paolo Guiotto: This family contains the intervals, so G contains intervals.
33:43:830Paolo Guiotto: of R… And this is clear, because if you are an interval, you are a Borisat.
33:50:860Paolo Guiotto: and the set X belongs to I is measure… is in this family of measurable sets.
33:59:400Paolo Guiotto: If E equals I is an interval.
34:04:960Paolo Guiotto: then the set X belongs to E is the set X belongs to the interval I.
34:13:130Paolo Guiotto: And that belongs to F, because…
34:19:969Paolo Guiotto: X is a random variable, which is just a way to say measurable function.
34:26:850Paolo Guiotto: Random variable, equivalent to measurable.
34:31:520Paolo Guiotto: function.
34:33:469Paolo Guiotto: And measurable function means that the set of omega for which X of omega belongs to i is in the sigma algebra F. So this family contain intervals.
34:45:239Paolo Guiotto: Then, step 2, the family G is a sigma algebra.
34:53:260Paolo Guiotto: So you have to prove that, it contains the empty set and the full space omega, the AR in G.
35:04:90Paolo Guiotto: And this is trivial. How do you get the empty set, here? Sorry, not the empty set, but the interior line.
35:19:900Paolo Guiotto: Yeah, because,
35:24:880Paolo Guiotto: Yeah, because if you take the set where X belongs to empty, what is this set? It's the empty set.
35:39:620Paolo Guiotto: And if you take X belongs to R, this is the full space omega. And since these two are in F, it means that MT and R are in the family
35:52:970Paolo Guiotto: G.
35:53:970Paolo Guiotto: Because the family G is the family of sets.
35:58:120Paolo Guiotto: for which the set X belongs to E is in F.
36:02:80Paolo Guiotto: So when I take empty set, X belongs to empty is the empty set which is in F. When I take E equal R, X belongs to R is the full omega, the full sample space, which is, of course, an element of F.
36:18:500Paolo Guiotto: Then you have to take… if you take an E that is in G, then E complementary is in G. Also, this…
36:26:640Paolo Guiotto: is easy, because if E is in G, you have that X belongs to E, is an event of the family F.
36:36:410Paolo Guiotto: But if it's a sigma algebra, right? So, when I do the complementary of this guy, X belongs to the complementary, the sigma algebra is closed with respect to this operation. So, also, this is an event.
36:50:960Paolo Guiotto: But that set is the same of X belongs to E complementary.
36:57:760Paolo Guiotto: Because if you are here, you have an omega for which X of omega cannot be any, otherwise you wouldn't be there.
37:08:710Paolo Guiotto: And this is exactly…
37:11:400Paolo Guiotto: saying that this is in F, and therefore E complementar is in G. And similarly, G is closed,
37:22:650Paolo Guiotto: by… Countable.
37:26:350Paolo Guiotto: unions.
37:29:830Paolo Guiotto: Okay?
37:32:10Paolo Guiotto: Now, this means that G is a sigma algebra, which is the second step, so you know that G contains all intervals. It is a sigma algebra, and therefore, G will contain the sigma algebra generated by intervals.
37:52:510Paolo Guiotto: Which is, liberal Sigma algebra.
37:56:740Paolo Guiotto: So, the Boreal sigma algebra is contained in G, and this means that, in particular, for every
38:04:800Paolo Guiotto: for every Borel set, the set where X belongs to E is an event. It's a set to which we can assign a probability.
38:15:120Paolo Guiotto: So this implies that the first part of the statement is true. Then, if you define,
38:25:50Paolo Guiotto: mu of X, mu X of E. By definition, the probability that X belongs to E
38:33:170Paolo Guiotto: This is well-defined, because we know now that this is an event, so we can assign a probability to that set.
38:41:740Paolo Guiotto: So this is, well-defined.
38:48:760Paolo Guiotto: And it is a probability, because mu X of empty, we already said it is the probability that X belongs to empty, this will be 0, and mu X of r is the probability that X belongs to R.
39:07:950Paolo Guiotto: Now, this would be the probability of omega, so 1…
39:12:850Paolo Guiotto: And finally, the countable additivity. If you have a disjoint union, of, Ian here.
39:21:750Paolo Guiotto: you want to show that this is… the mu X of this is the sum of the mu X, but this means that it is the probability that X belongs to that disjoint union of the yen. Now, this means,
39:36:260Paolo Guiotto: If you want to see formally, this is the probability of the set of omegas, such that the value X of omega belongs into that disjoint union of the yen.
39:48:940Paolo Guiotto: But if you have an omega here.
39:51:890Paolo Guiotto: It means that it is in the disjoint union, so it is in the union, it is in one of the yen, and since the union is disjoint, it is exactly in only one of the yen.
40:03:140Paolo Guiotto: So this is the same of the disjoint union of the set of omegas where X of omega is in a single EN, you see?
40:15:260Paolo Guiotto: But that P is a probability, so you can say that P of a disjoint union is the sum of the P's, so probability that
40:24:330Paolo Guiotto: X belongs to EN, and these, by definition, are the mu X of EN.
40:33:690Paolo Guiotto: And so we have the countable additivity.
40:36:770Paolo Guiotto: So, MUX… is countably.
40:42:710Paolo Guiotto: additive.
40:44:740Paolo Guiotto: And this finishes the proof.
40:47:120Paolo Guiotto: Now, this function new x is called also low, or distribution of X.
40:55:50Paolo Guiotto: Okay, so, definition…
41:01:550Paolo Guiotto: MUX, is called…
41:08:410Paolo Guiotto: low.
41:14:250Paolo Guiotto: distribution.
41:19:380Paolo Guiotto: Auvix.
41:24:90Paolo Guiotto: Let's see, some example.
41:29:660Paolo Guiotto: Soft.
41:30:890Paolo Guiotto: Example… the first examples are trivial, more or less, because we do not dispose yet of
41:37:890Paolo Guiotto: Strong calculations. So, let's consider first the simplest possible random variable, a constant random variable.
41:47:120Paolo Guiotto: So, constant…
41:54:10Paolo Guiotto: random variable.
41:55:970Paolo Guiotto: Constant means that I have an X, which is constantly equal to some value, say, X0.
42:05:460Paolo Guiotto: on the probability space, omega FP,
42:12:910Paolo Guiotto: Now, what is the mu X of this? According to the definition, mu X of set E is the probability that X belongs to the set E, right? That's the definition of mu X.
42:29:810Paolo Guiotto: Now, this X is just equal to X0, so this is the probability that X0 belongs to E.
42:42:20Paolo Guiotto: Now, this is an all-or-nothing event, because you have basically two possibilities. Either your X0 is in the set E,
42:56:70Paolo Guiotto: So that's a true reality, no?
42:59:920Paolo Guiotto: And this means that whatever is E, that is, well, if you want, this is formally.
43:07:540Paolo Guiotto: the probability, we are doing probability on omega, probability of set of omega C in capital omega, such that X0 belongs to E. Now, suppose that that X0 is in E, okay?
43:23:210Paolo Guiotto: then what is the set of omegas for which X0 in E is true? Well, every omega. So, if,
43:32:230Paolo Guiotto: let's say we have this alternative. If X0 is in E, this set here
43:38:990Paolo Guiotto: Is all the elements of the set omega, because that condition is independent of the omega, no?
43:46:340Paolo Guiotto: It's a condition that does not depend on the omega, because that X0 is a constant. So I have probability of omega here, and that's equal to 1.
43:56:750Paolo Guiotto: While, if X0 is not in E,
44:00:680Paolo Guiotto: What happens? Well, there is no omega that can make X0 in E true, because that condition is always false.
44:09:350Paolo Guiotto: So this means that the set of omega in that case is just empty, and therefore the probability will be zero.
44:16:790Paolo Guiotto: So mu X of E is 1 when X0 is in E, and 0 when X0 is not in E.
44:25:620Paolo Guiotto: This is the direct delta measure, so it's the delta X0P.
44:33:700Paolo Guiotto: So we may say that in this particular case, the law of X reduces to a delta
44:42:890Paolo Guiotto: Dirac delta centered, concentrated at point X0, which is the point where the variable is concentrated.
44:52:30Paolo Guiotto: So, you see, you should start noticing this.
44:57:430Paolo Guiotto: That we will repeat many times. We don't know what is the space of ego here, you see? It can be whatever.
45:07:590Paolo Guiotto: A space of spaghetti and mozzarella and pizza. But, now, the low is a probability on R, so that's a sort of more concrete object.
45:19:140Paolo Guiotto: Even if it is not particularly easy to handle, because it is a measure. In fact, there will be, let's say, easier tools than this one.
45:30:360Paolo Guiotto: But this is the first, let's say, simplification. It says that the law at the end is this direct delta distribution. Now, let's take a non-constant example.
45:46:930Paolo Guiotto: We call this, example.
45:52:930Paolo Guiotto: So this, the first non-constant random variable, we may imagine, if a constant takes one single variable, the non-constant can take two values, okay? So this is called, Bernoulli
46:09:600Paolo Guiotto: a Bernoulli, random variable. A random variable that takes two values is called the Bernoulli random variable.
46:19:420Paolo Guiotto: Mmm… so, assume that we have X of omega, equal, certain value, X0, well, X1.
46:37:350Paolo Guiotto: on, for omega in some event. Well, let's use letter… F1… X2 for omega in… F2…
46:53:20Paolo Guiotto: Where these two…
46:55:160Paolo Guiotto: are, of course, we suppose that the two values are different, so the two sets cannot be, cannot have common points, and, so they are disjoint, and the union makes all the
47:08:480Paolo Guiotto: omega, okay? So a Bernoulli random variable is something like this.
47:12:760Paolo Guiotto: It takes, value X1 on a certain event F1,
47:19:300Paolo Guiotto: it, it takes values to run a certain other event. Now, what is the, the law of this,
47:27:940Paolo Guiotto: random variable. So take a set E, we have to compute the probability that X belongs to E.
47:38:50Paolo Guiotto: Now, again, this is the probability of the set of omegas
47:43:730Paolo Guiotto: in the sample space, for which we have X of omega, is in E.
47:51:710Paolo Guiotto: Now, this X of omega can take only two values, no? Which are, the value X1,
48:01:180Paolo Guiotto: when omega is in F1, the value X2 when omega is in F2. So that X of omega can only be equal to X1 or X2.
48:14:750Paolo Guiotto: And therefore, we will have a condition like X1 is in E, or X2 is in E, so it's similar to the previous one. But that condition, the important fact is that it is independent of omega.
48:28:900Paolo Guiotto: Okay? So we may say that, okay, how we can discuss this. Well, since this is split into two subsets, let's do this. So we take the omega,
48:44:120Paolo Guiotto: in F1, Such that X of omega belongs to E.
48:52:250Paolo Guiotto: We can make this disjoint union with the omega in F2, such that X of omega is in E. You see, because we said that
49:04:280Paolo Guiotto: this… this joint union yields the full sample space omega, no? So I can say, take this omega and divide in two, no? This is the sample space omega on a part F1, the random variable X,
49:23:680Paolo Guiotto: yields value, here we are on the real line, yields value X1, I don't know, here. On the other part, F2, the value is X2 different.
49:35:800Paolo Guiotto: So I'm saying, let's split the domain in two parts.
49:40:190Paolo Guiotto: No? The part F1 and F2, they… they…
49:43:370Paolo Guiotto: combined, they give all omega, and let's split this set here into two subsets, disjoint. Therefore, the probability of their union will be the sum of the probabilities. So we will have probability omega in F1, in this case, X of omega.
50:01:770Paolo Guiotto: is equal to X1,
50:04:40Paolo Guiotto: And so we have that this is X1 belongs to E, plus the probability that omega is in F2,
50:13:690Paolo Guiotto: Such that, in this case, X of omega is X2, that belongs to you.
50:20:800Paolo Guiotto: Now, what could happen to these… two probabilities.
50:25:880Paolo Guiotto: Let's think about, no?
50:27:890Paolo Guiotto: You see that everything depends on these two conditions. X1 is in E, X2 is in E, which are similar to the previous discussion, no?
50:37:300Paolo Guiotto: They are independent of omega, so they are either true or false.
50:42:380Paolo Guiotto: conditions.
50:43:690Paolo Guiotto: So, they are true, for example, X1 in E is true when X1 belongs to the set E, otherwise it is false, and similarly for the second case. So we may have, basically, four possibilities.
50:57:820Paolo Guiotto: Set E does not contain any of the two.
51:02:250Paolo Guiotto: So X1 is not in E, and also X2. So the two conditions are false.
51:08:210Paolo Guiotto: Or, the set E contains only one of these two points.
51:13:510Paolo Guiotto: So, only one of these two conditions is true. Or, again, the set E contains both, so the two conditions are true. So, we have to distinguish four situations, four cases. So, case X1 and X2 are not
51:32:130Paolo Guiotto: in E, case X1 is in E, but X2 is not in E, case X1 is not in E, and X2 is in E. And finally, case X1 and X2 are both in E.
51:51:840Paolo Guiotto: Now, first case, if X1 and X2 are not in E, it means that the two blue conditions are false. So there are no omega for which X1 is in E. So that's the probability of the empty set.
52:07:720Paolo Guiotto: plus… and also there is no omega in F2 for which X2 belongs to E, because that's… that's false. So, this is a probability of empty plus probability of empty, so this is 0.
52:21:270Paolo Guiotto: Now, in the second case, X1 is in E. So the set of omega of F1, for which X1 is in E, all the omega of F1, okay? Because the condition X1 is in E is, in this case, is true, okay? While in this case, this is 4.
52:40:650Paolo Guiotto: So, this means that I have a probability of… what is the set of omega in F1 such that X1 is in E? All the omega of F1, so probability of F1, plus what is the probability of omega in F2 for which X2 is in E? There is no omega here, because X2 is not in E, so that's…
52:59:820Paolo Guiotto: will be empty. Probability of empty, and they get probability of F1.
53:06:530Paolo Guiotto: The opposite case, we have, similarly, that in the first set we have empty, in the second we have F2.
53:14:170Paolo Guiotto: So at the end, we have probability of F2.
53:19:270Paolo Guiotto: And the final one, we have that both these two conditions are true, no? This and also this.
53:28:00Paolo Guiotto: So when you say, what are the omegas for which X1 is in E? X1 is in E, so each omega of F1, of course, so you have probability of F1,
53:38:660Paolo Guiotto: plus probability of F2.
53:41:910Paolo Guiotto: And this one, since the two are…
53:45:170Paolo Guiotto: If this joint union is omega, that's equal to 1.
53:49:290Paolo Guiotto: So we have this. We may also write this in this form, that if we call, for example, P, the probability that… of the set F1, which is the probability where the random variable X is equal to X1, as we said.
54:08:910Paolo Guiotto: The other one, the probability… if we call Q the probability of the set F2, which is the probability where the random variable X is equal to X2,
54:20:840Paolo Guiotto: Well, since the random variable X can be only one or the other, this event is the complementary of the other, so this will be 1 minus P.
54:31:40Paolo Guiotto: So at the end, we can say that the probability is 0 when X1 and X2 are not in E.
54:41:130Paolo Guiotto: It is, say, P when X1 is in E.
54:46:430Paolo Guiotto: But X2 is not in there.
54:49:530Paolo Guiotto: Eve… It is 1 minus P when X1 is not in E, and X2 is in E.
55:00:280Paolo Guiotto: And finally, it is 1 when both X1, X2, In… il.
55:06:870Paolo Guiotto: So that's the… the… not delta, the mu X.
55:12:880Paolo Guiotto: of the set B.
55:16:490Paolo Guiotto: You'll notice again that at the end, this is the low of the random variable.
55:24:80Paolo Guiotto: If you look at the low, you don't have to know where the random variable… you remind this story. Now, the random variable is X1 on a certain set event F1, it is X2 on another event, F2, in such a way that
55:39:590Paolo Guiotto: made called the Omega together.
55:42:200Paolo Guiotto: But you need just to know what? This number here, which is the probability where the random variable is X1,
55:52:890Paolo Guiotto: And, because the other one, it's just 1 minus B.
55:57:110Paolo Guiotto: And the two values that the random variable takes.
56:01:840Paolo Guiotto: The particular case is the case…
56:06:460Paolo Guiotto: particular case, which is the case used in most of the applications, it is you have X random variable that takes two particular values, values 0 and 1. It is used to do a test, true or false, how to say, huh?
56:23:490Paolo Guiotto: So, random variable, taking…
56:31:130Paolo Guiotto: Values.
56:34:70Paolo Guiotto: 0 and 1. So we may say that P is the probability that X is equal… well, conventionally, the letter P is reserved for X equals 1, so 1 minus P will be the probability that X is 0.
56:52:680Paolo Guiotto: In this case, the mu X… of P… ESA.
56:58:970Paolo Guiotto: So, the two values, you see that this is, this is, it doesn't matter the order, let's say this is X1, this is X2.
57:09:950Paolo Guiotto: So we have that it is 0 when 0, 1 are not in E. It is P when 1 is in E, but 0 is not in E.
57:23:520Paolo Guiotto: It is 1 minus P.
57:26:660Paolo Guiotto: when one is not in E, and 0 is in E,
57:31:200Paolo Guiotto: And finally, it is equal to 1 when both 0 and 1 are elite.
57:37:420Paolo Guiotto: This is the… low of a Bernoulli random variable.
57:46:320Paolo Guiotto: Okay.
57:48:10Paolo Guiotto: We will introduce a more complex example later, because handling a low, it's better than handling a probability, because on…
58:01:720Paolo Guiotto: The measure mu X, is a measure on a space, is the real line
58:08:110Paolo Guiotto: with sets that we know, the Borel sets.
58:11:870Paolo Guiotto: While the probability PE might be defined on a set of
58:17:00Paolo Guiotto: Much more complicated, without any metric structure, so it could be worse.
58:24:690Paolo Guiotto: Well, at the end, it is basically possible to do any calculation on the random variable just by using the law of X.
58:33:460Paolo Guiotto: For example, -Oh.
58:37:990Paolo Guiotto: D… low.
58:42:450Paolo Guiotto: off.
58:44:50Paolo Guiotto: The random variable accent.
58:48:930Paolo Guiotto: are loads.
58:52:400Paolo Guiotto: tooth.
58:53:770Paolo Guiotto: Calibrate.
58:57:00Paolo Guiotto: And it… relevant… Oh, entiti…
59:09:390Paolo Guiotto: associated… To a random value.
59:15:320Paolo Guiotto: So we already have an example of this. For example.
59:21:380Paolo Guiotto: you want to compute the significant probability. Probability that X does something can be always written in the form probability that X belongs to a certain set E, no? That's the mu X of E. Suppose that you have this measure.
59:39:270Paolo Guiotto: No? You can compute these probabilities just by doing this.
59:45:390Paolo Guiotto: Another example, important example, is, for example, what about…
59:54:320Paolo Guiotto: the expected value of X.
59:57:730Paolo Guiotto: Provided that… X is in L1.
00:04:720Paolo Guiotto: Can we compute the expected value of X by using the law of X?
00:11:190Paolo Guiotto: And how it works. Well, we can do, and we can understand what is the argument by looking inside this identity here.
00:21:860Paolo Guiotto: Because this identity can be written in this way. Now.
00:27:820Paolo Guiotto: A probability can be always written as any measure, as an integral, no?
00:35:100Paolo Guiotto: So, the idea is that for a measure mu, mu of a set S, I use this notation, that will be applied to both sides. Mu of S is the integral on S of 1 in mu, okay?
00:51:930Paolo Guiotto: So we're going to use this factor. So, how can I see the probability of the event X belongs to E?
01:00:460Paolo Guiotto: Well.
01:01:260Paolo Guiotto: I could see this as… this is the event, the set of omega, such that X belongs to E. So a possibility would be, let's write this as the integral on the set where X belongs to E of 1 in the measure P, no? This is a possibility.
01:22:690Paolo Guiotto: But also, I could put this into the integral through an indicator, say that this is the same of the integral on omega, and now I put a function which is 1, exactly when X belongs to E.
01:37:920Paolo Guiotto: and 0 when X is not in E.
01:41:350Paolo Guiotto: So this is the sale.
01:44:750Paolo Guiotto: Okay, we can do a little bit better, because we can say this is the integral on omega of indicator of the set E evaluated on X. Because, think about, when this quantity is 1,
02:01:940Paolo Guiotto: This is one when X of omega belongs to the set E. It's the same thing.
02:09:690Paolo Guiotto: Okay, so I have that this can be written this way. On the other… in the other side, I have mu X of E that can be, again, written as integral on e of 1 in the measure mu X.
02:23:780Paolo Guiotto: But also, carrying this into the integral, I could say this is the integral on the real liner.
02:30:590Paolo Guiotto: of the indicator of set E,
02:34:200Paolo Guiotto: in the measure of mu X.
02:37:880Paolo Guiotto: So, the definition of law of a random variable can be written in this little bit exaggerated form.
02:47:230Paolo Guiotto: So…
02:52:880Paolo Guiotto: mu-X.
02:54:250Paolo Guiotto: P is D.
02:56:790Paolo Guiotto: low.
02:58:740Paolo Guiotto: Off.
02:59:680Paolo Guiotto: X, if and only if I have this identity.
03:04:370Paolo Guiotto: the integral on R of the indicator of the set E in the measure muax, in the low of axel.
03:12:780Paolo Guiotto: is equal to the integral on omega of the indicator of E, but
03:18:250Paolo Guiotto: Evaluated, composed with X with respect to the probability P.
03:23:740Paolo Guiotto: So if you want to see the variables, in a little bit, say.
03:28:750Paolo Guiotto: extremely precise notation. This at left is the integral of this function, indicator of X. Let's call little x the variable here. This X
03:42:240Paolo Guiotto: is a real variable, so respect to the measure mu X, that is a measure on the real line. On the right-hand side, we have an integral in omega of indicator E of X. X is a function of omega, so to be precise, I should write X of omega.
04:01:740Paolo Guiotto: DP here, and here we are integrating in omega. We will never use anymore this second notation, because the first one is a little bit,
04:12:510Paolo Guiotto: better, But here you see that,
04:18:230Paolo Guiotto: In fact, if you take omega, you compute X omega, X omega belongs to E, you get 1. So it's the same of the indicator, it's the same meaning of this indicator here.
04:31:260Paolo Guiotto: Okay? Because that is one…
04:33:340Paolo Guiotto: on the set of omegas, because that's, again, a function of omega, it is 1 on the set of omegas for which that omega belongs to the set of omegas in capital omega, where X of omega is in E, no?
04:48:830Paolo Guiotto: This would be the, a bit cumbersome notation.
04:54:250Paolo Guiotto: Okay, so, however, the definition of mu X is equivalent to this relation.
05:01:320Paolo Guiotto: Now, I want to transform this into a relation where I read and write the integral of X in DP, so the expected value of X, and that left something else.
05:14:630Paolo Guiotto: Well, how can we done this? The first remark is that, okay, if this is valid for an indicator of a set E,
05:23:800Paolo Guiotto: by linearity, I can do something more. So, if you take a linear combination.
05:30:580Paolo Guiotto: something like CJ, indicator of EJ,
05:36:60Paolo Guiotto: and you consider this, this is, I remind you.
05:40:460Paolo Guiotto: A function that takes a finite number of values, so this is a simple function. Simple function.
05:51:50Paolo Guiotto: Now, by linearity, you can… this is a finite sum, so let's say J from 1 to, you know, Candelena.
05:58:900Paolo Guiotto: You can carry outside, from this integral, the sum, you can carry outside also the coefficients, which are constants, so you have sum J equal 1 to N, CJ integral on R, indicator of EJ,
06:17:330Paolo Guiotto: VMUXA.
06:18:960Paolo Guiotto: Now, these are these quantities we have here.
06:22:820Paolo Guiotto: And they coincide with these other quantities. So we replace, at this stage, these integrals with the integral on omega of indicator set EJ, evaluated on X in the probability thing.
06:40:30Paolo Guiotto: And now we do the opposite operation, so we return this coefficients inside, and we carry back the sum inside. So we have integral on omega sum j going from 1 to capital N,
06:57:670Paolo Guiotto: CJ, indicator EJ, evaluated at X, DP.
07:04:550Paolo Guiotto: Now, if you write in red this,
07:07:330Paolo Guiotto: So look at this function, sumJ1 to N, CJ…
07:13:00Paolo Guiotto: is the indicator set EJ, this is a function of X, and this is a simple function, let's call it S .
07:23:290Paolo Guiotto: It's the same that you see here.
07:26:570Paolo Guiotto: It is S evaluated on capital X. So we have this
07:33:90Paolo Guiotto: extension of this formula, let's call it 1.
07:38:90Paolo Guiotto: This yields formula 2, integral on R of S d mu X,
07:46:540Paolo Guiotto: equal to integral on omega of S of capital X DP,
07:54:30Paolo Guiotto: for every S, which is just a simple function.
08:00:290Paolo Guiotto: Simple, function.
08:07:50Paolo Guiotto: And now we extend, again, this formula.
08:11:50Paolo Guiotto: Because we remind that if we have… we… we proved somewhere
08:17:69Paolo Guiotto: It was a long time ago, that a remarkable fact
08:25:740Paolo Guiotto: This solves for any measure, okay? If you have a function f, He's, mu, measurable.
08:38:279Paolo Guiotto: So, it is measurable with respect to the measurement. Then, and it is positive.
08:44:990Paolo Guiotto: This is important. There exists a sequence SN of simple functions.
09:00:640Paolo Guiotto: which is an increasing sequence, so this means that SN plus 1 is above SN.
09:09:420Paolo Guiotto: And such that this sequence XN converges pointwise everywhere to your function f.
09:17:720Paolo Guiotto: We even built a formula for that. It was in the first part, but just to remind you where it was…
09:30:170Paolo Guiotto: Nope.
09:34:319Paolo Guiotto: Maybe you, you may, you will remind?
09:39:470Paolo Guiotto: Of this, yes, this is the construction.
09:43:130Paolo Guiotto: of the integral.
09:47:370Paolo Guiotto: Well, follow me.
09:50:920Paolo Guiotto: It is here where we built the simple functions. So, this was the definition of integral, but…
09:59:530Paolo Guiotto: Or maybe the functions are in the next one.
10:07:310Paolo Guiotto: Here we are. These are the functions, sir.
10:10:190Paolo Guiotto: These are… this construction has nothing to do with the LeBague measure. It was on a generic measure space. We took a positive measurable function.
10:21:550Paolo Guiotto: And we built these functions as N, and they have this property. They are an increasing sequence of functions, and pointwise convergent at each point.
10:35:60Paolo Guiotto: not almost everywhere convergent. No, it's everywhere convergent to the function F. So it's that, it's that fact that we are reminded here. So what we can say that, is, we plug this into the formula, so…
10:54:160Paolo Guiotto: we have the integral on R of Sn d mu X. This will be the integral on omega of SNX
11:05:150Paolo Guiotto: in the probability P.
11:07:590Paolo Guiotto: For every annual.
11:09:340Paolo Guiotto: I just plug the formula to here. But now what I do is I do the limit.
11:16:890Paolo Guiotto: I can do the limit because this is a sequence of a positive function which is increasing pointwise to F, and so I can use the monotone convergence theorem, so monotone…
11:31:900Paolo Guiotto: convergence.
11:33:430Paolo Guiotto: This now will go to the integral on R of F d mu X, and this will be equal to the integral on omega to F of F of capital X DP.
11:50:560Paolo Guiotto: They sold, sir, for every… F, positive.
11:55:690Paolo Guiotto: and measurable respect to the measure we are considering here, so, which is the mu X.
12:03:880Paolo Guiotto: mu X measurable, but mu X measurable means that measure is a Boreal, measure, so, so measurable.
12:17:950Paolo Guiotto: With respect to… the Boray class.
12:26:930Paolo Guiotto: And this is the formula number 3. Now we have extended that relation to a generic F, which is positive and borel measurable. And now the last step is Formula 3 as a final extension, which is 4, that you may imagine.
12:47:280Paolo Guiotto: Now, by taking positive and negative part, we will have that this formula becomes true
12:53:550Paolo Guiotto: Provided the function f is integral.
13:02:40Paolo Guiotto: So, if F is in L1,
13:07:90Paolo Guiotto: R, the Borel class is the sigma algebra, and the measure is the measure nuance.
13:16:770Paolo Guiotto: And, so this is important because it says that,
13:20:710Paolo Guiotto: So, now let's return to the probabilistic notations. That integral is an expectation. So, if you want to compute the expectation of any function of your random variable x.
13:35:790Paolo Guiotto: Okay? Now, the point is that the integration on this side could be complicated, because maybe the space is not a finite dimensional space, and the integration is the measure P, the probability is complex, you don't know how to handle. I'm thinking the case when we have, for example, the Brownian motion, the omega is an infinite dimensional space.
14:00:750Paolo Guiotto: these are paths, no, continuous functions, and the probability is the linear measure, which is a probability on a space of paths. It's something that we can even…
14:12:630Paolo Guiotto: Handle this, but it says you can reduce this to a fin-dimensional stuff, a one-dimensional stuff, provided you know the law of the random variable.
14:22:930Paolo Guiotto: And this is the integral. So, in particular.
14:27:930Paolo Guiotto: For example, you want to compute the expected value of X,
14:33:730Paolo Guiotto: You can do, because it's, like, if you take F of X equal X, so it is the integral in the real line of X in the low d mu
14:45:620Paolo Guiotto: Of X.
14:46:830Paolo Guiotto: I'm not saying that this is trivial, because, of course, the measure new might be… might be complicated, so it's not at all,
15:00:50Paolo Guiotto: A simple, a simpler.
15:04:00Paolo Guiotto: Factor?
15:09:200Paolo Guiotto: But it might be, it might be easier.
15:22:260Paolo Guiotto: Okay.
15:28:10Paolo Guiotto: So…
15:29:270Paolo Guiotto: This is the concept of flow. The next, it is a… a father, it's a new object, which is a,
15:40:960Paolo Guiotto: again, a more concrete object with respect to a measure. It is a function, and this is the so-called
15:50:10Paolo Guiotto: CDF, cumulative.
15:56:320Paolo Guiotto: distribution…
16:02:930Paolo Guiotto: functions.
16:07:310Paolo Guiotto: So, what is this?
16:09:20Paolo Guiotto: Well, this is a particular case of the law of X.
16:16:140Paolo Guiotto: So, consider the… this, you know that the law of action, given… a random variable X.
16:32:470Paolo Guiotto: mu of X is a boreal… measure.
16:39:730Paolo Guiotto: So, in particular, you can… You can compute the measure of any interval.
16:47:820Paolo Guiotto: Now, we consider the measure of a particular interval of this form minus infinity to some value X. Be careful, because here it is very important
17:01:530Paolo Guiotto: The, the shape of this interval.
17:04:920Paolo Guiotto: But, of course, you go to, up to minus infinity, at the left turn, at right, you go up to X included, okay?
17:13:790Paolo Guiotto: Now, this means that it's the same of the probability that X belongs to the interval minus infinity to little x, or better.
17:25:380Paolo Guiotto: the probability that X is less or equal than little X.
17:32:40Paolo Guiotto: Which is something that, has some,
17:35:720Paolo Guiotto: Meaning, now we want to know what is the probability that a random variable is smaller than a fixed value X.
17:43:120Paolo Guiotto: Now, we define this as a function f sub X, of the little x as a numerical function, because as you can see, when you, take any X real, this X is real.
18:00:180Paolo Guiotto: You define properly a function.
18:03:500Paolo Guiotto: Now, this F is what is called FX is called… Cdf.
18:16:150Paolo Guiotto: Cumulative distribution function of Excellent.
18:22:980Paolo Guiotto: Now, basically, what it will turn out is that to each X, there is, of course, a CDF, an FX, but
18:33:590Paolo Guiotto: also, in some sense, a vice versa holds, provided we characterize properly what are the key features of this function. So let's see the main properties of this proposition.
18:49:650Paolo Guiotto: So, the… CDF, FX of X.
18:59:800Paolo Guiotto: verifies.
19:03:290Paolo Guiotto: The… Following…
19:10:590Paolo Guiotto: He… properties.
19:18:220Paolo Guiotto: So, the first one is, almost evident.
19:23:140Paolo Guiotto: When you increase X, what happens? Well, you see that if you increase X, that segment minus infinity 2x becomes bigger.
19:32:840Paolo Guiotto: And since the set is bigger, the measure increases, and so we have that the function fx is increasing in X. So this means that FX of, say.
19:46:710Paolo Guiotto: X is less or equal than FX of Y for every X less or equal than Y. I'm not saying strictly increasing, because this one's being general are true, just increasing.
20:03:230Paolo Guiotto: The second property is what happens at infinitives. Imagine that you go down X to minus infinity.
20:11:710Paolo Guiotto: Intuitively, the offline to minus infinity to X shrinks down to nothing.
20:18:880Paolo Guiotto: Because if X goes to minus infinity.
20:21:650Paolo Guiotto: you should have the numbers which are less or equal than X with X going to minus infinity. Nothing. So, you expect that when you go at minus infinity, I will shortly write this, that means the limit
20:36:370Paolo Guiotto: for x going to minus infinity of FX of little x, this will be 0.
20:44:490Paolo Guiotto: While, and this is a bit more evident, when you go on the other side at plus infinity, so you compute the limit when x goes to plus infinity of F sub X of little x.
21:00:60Paolo Guiotto: But in this case, if you can put little x equal to plus infinity, it means
21:05:90Paolo Guiotto: The probability that your variable is less or equal than plus infinity.
21:09:670Paolo Guiotto: It's 1, no? Because…
21:13:800Paolo Guiotto: Third property is a continuity property. Third and fourth, actually, they are a unique property. So, FX is… the function in general is not continuous, but we can say that it is right continuous.
21:32:150Paolo Guiotto: at every point X in R. So, right continuous means that when you do… continuity means that when you do the limit, at some point, you get the value of the function. That's the continuity. Right continuity means that if you do the limit from the right, so say that you take Y, you go to X,
21:51:590Paolo Guiotto: from the right, so this is the real line, this is X. You move Y to X from this side, the FX of Y
22:01:990Paolo Guiotto: goes exactly to FX of X.
22:08:840Paolo Guiotto: While, from the left, we do not have the same, otherwise we would have continuity, but FX has left
22:21:370Paolo Guiotto: limits.
22:24:340Paolo Guiotto: for every X real. So in this case, when we move Y to X from the left, the limit
22:32:710Paolo Guiotto: for Y going to X minus, this time, of FXY. This can be said that it exists.
22:42:610Paolo Guiotto: And moreover, we can just… Give a little bit of…
22:48:400Paolo Guiotto: Information which is less or equal than the value of the function at point L.
22:55:80Paolo Guiotto: So these are the characteristic properties of any
22:59:920Paolo Guiotto: CDF, Cumulative Distribution Function. We can have an idea, so what is the shape, the typical shape of this, of this type of function? So, we are in the real line.
23:13:250Paolo Guiotto: The first remark is that since this function, by definition, is a probability, or if you want the low, the low is a probability, as we said.
23:24:240Paolo Guiotto: So, the values of this function will be positive and be less than 1. So, the graph of the function will be between these two lines.
23:37:140Paolo Guiotto: Above 0, below 1.
23:40:100Paolo Guiotto: Then…
23:41:440Paolo Guiotto: Property number 1 says the function is increasing, so we see an increasing function. That property number 2 at minus infinity goes down to zero. So we have to imagine that our function typically goes to zero when we go far
23:59:00Paolo Guiotto: To the left.
24:00:520Paolo Guiotto: And when you go to plus infinity, it goes to 1.
24:04:410Paolo Guiotto: It does not mean that it is not 1. As you will see, it can be, well, 1 and 0, okay?
24:10:790Paolo Guiotto: it is increasing, then if you take any point X, you have that it is right continuous, so you have to imagine that here there is a value, which is the value at point X, FX of X, and if you move to X from the right.
24:29:650Paolo Guiotto: The function is increasing, so it's decreasing going from right to left, no? And it is continuous, so you have to imagine something like this.
24:39:530Paolo Guiotto: So I don't know what happens here, but it's an increasing function.
24:43:330Paolo Guiotto: At the same point, you might not have the continuity, maybe, so you can have that it is continuous.
24:50:570Paolo Guiotto: Or, if it is not continuous, there is a jump. The jump is a jump, let's say, where the left value is smaller.
25:00:560Paolo Guiotto: the left limit that you see here is always smaller than the value of F at point X, so this will be more or less the shape of this VF. So, there might be several jumps, so something like this, so another jumps down here.
25:18:720Paolo Guiotto: Maybe like this, and this is constant equal to zero, for example, no?
25:25:860Paolo Guiotto: So this is typically what is a plot of SDF for this.
25:33:400Paolo Guiotto: Well, let's see, the proof is not,
25:37:220Paolo Guiotto: complicated, and it uses what we learned about measures in general. So, number one is evidence, because if you take FX of X,
25:48:330Paolo Guiotto: So take X less or equal than Y. This is… let's write in terms of probability, if you want. It is the probability that X is less or equal than X.
26:00:250Paolo Guiotto: What is the relation between this probability and this one? The probability that X is less than Y?
26:06:810Paolo Guiotto: It is clear that if little x is less than little y, if you are smaller than little x, you are smaller also than little y. So this event is contained into this one.
26:20:400Paolo Guiotto: And therefore, the second one will be bigger, and it will have a bigger probability. So this will be less or equal than this. It's because of the monotonicity of any measure, and that's exactly FX at point Y.
26:34:540Paolo Guiotto: So, number two, the limits. At minus infinity, let's do that minus infinity, and similarly you can do it plus infinity.
26:44:40Paolo Guiotto: Now, if you want to compute the limit when x goes to minus infinity, you have, of the function fx of x, you have to do the limit when x goes to minus infinity.
26:59:210Paolo Guiotto: Of, well, let's look at this, by using the mu axis in different… you can write the proof with the low or with the… with the probability piece.
27:15:960Paolo Guiotto: Now, what is the idea? You have a limit of measures. What these sets do? Well.
27:22:900Paolo Guiotto: you imagine that if you move X to minus infinity, these sets are reducing, no? Because this is an offline from minus infinity to X. So if you take a 2X, say, X1, which is less than X2, you see the offline from minus infinity to X1 is smaller than the offline from minus infinity
27:46:130Paolo Guiotto: to X2, and that's clear.
27:48:600Paolo Guiotto: So, if these X are somehow
27:53:10Paolo Guiotto: decreasing down to minus infinity, these sets, the offlines, are decreasing down, so this should remind of a continuity property.
28:02:730Paolo Guiotto: But continuity demands a discrete set, so a discrete number of sets, so I should have something like a countable number of sets. So, how can I do this? Well, the typical technical argument is the following.
28:18:500Paolo Guiotto: To check this, we proceed by sequences. So, let's prove…
28:26:990Paolo Guiotto: bet.
28:30:70Paolo Guiotto: for every sequence XN, such that XN goes Down to minus infinity.
28:40:360Paolo Guiotto: Then, the limit in N, when n goes to plus infinity.
28:47:220Paolo Guiotto: of mu X of minus infinity to Xn.
28:54:190Paolo Guiotto: N is the index, so it goes to plus infinity, but Xn is going down to minus infinity. This is zero.
29:01:330Paolo Guiotto: Okay, so call this set SN.
29:05:390Paolo Guiotto: Since the sequence XN is decreasing, I have that SN plus 1, as in the figure here, will be contained into SN. So the sequence of sets Sn will be decreasing.
29:23:130Paolo Guiotto: Okay?
29:24:740Paolo Guiotto: Moreover, mu X is a probability measure.
29:31:830Paolo Guiotto: So, continuity… from… above.
29:39:670Paolo Guiotto: applies…
29:45:290Paolo Guiotto: And therefore, we can say that the limit of the measures, The continuity says that,
29:55:280Paolo Guiotto: when you have a sequence of sets going down to something, okay, so decreasing sequence, the limit of the measure is the measure of the limit of the… which is, in this case…
30:10:360Paolo Guiotto: intersection of all these sets, minus infinity to Xn.
30:18:100Paolo Guiotto: So this is the continuity from… above.
30:27:200Paolo Guiotto: Okay? But now, what is that intersection? Well, since the sequence XN is going down, no, X0, X1, etc.
30:38:760Paolo Guiotto: XN, this is moving to minus infinity.
30:43:20Paolo Guiotto: The intersection of this, is,
30:48:970Paolo Guiotto: the intersection of these sets is empty. Why? Because if an X belongs to that intersection.
30:59:300Paolo Guiotto: If it is in the intersection, it is in all of them, so X must be less or equal than XN for every N.
31:06:980Paolo Guiotto: But if you are less or equal than Xn for every n, and this guy goes to minus infinity, you must be less than minus infinity, which is impossible. So this means that, X should be less or equal than minus infinity, impossible.
31:23:890Paolo Guiotto: So this means that the intersection is empty, and therefore, the probability is zero.
31:31:340Paolo Guiotto: And this means that we proved that
31:34:350Paolo Guiotto: No matter how we take this sequence that goes down to minus infinity, the limit of the measures of this X is 0.
31:43:210Paolo Guiotto: And this means that the full limit will be zero.
31:46:720Paolo Guiotto: So, from this, limiter, for x going to minus infinity of fx of x equal to 0.
31:56:480Paolo Guiotto: Well, it's the same, so I invite you to do the proof. Prove the second limit, prove that FX at plus infinity is equal to 1. This means the limit at plus infinity is equal to 1.
32:11:890Paolo Guiotto: And similarly, you have the same for the left and the right limit. It's a similar idea, okay?
32:18:590Paolo Guiotto: 3 is similar.
32:25:10Paolo Guiotto: So we, we stopped just here. Okay, that's all for today.
32:30:750Paolo Guiotto: And see you tomorrow. I will publish soon the notes. I'm still updating, so I need a couple of days.
32:41:500Paolo Guiotto: again.
32:42:830Paolo Guiotto: Okay.
32:44:520Paolo Guiotto: Let's stop the recording.