The Affair Nifera for Collective Transformation has demanded for a youth presidential aspirant towards the forthcoming general election from all progressive Congress, the OPC, in the country. This was made known by the director general of the group, Tolua Jai, during the Southwest Benefit Conference held at the Premier Hotel, Ibadon, or your state. He explained that the aspirant should have vibrant traits which the youth possessed. What we are saying is that it's time for total handover to the youth. When we talk about youth, we talk of the people that are called the independence in the economy. Those that are meant to be running the affairs of the states that have been denied for quite four decades ago. People that are meant to develop the nation with their intellect, their energy, with their sincerity. Speaking at the event, the governor of Kogi State, Yahya Bello, represented by head of Kogi Blueprint, Moses Okezie, highlights the role of youth in political affairs in the country. Elders have done their best. I can tell you, we wouldn't be where we are as a nation. The good things that we can say about this country, we cannot say without the efforts of the elders. But even they themselves have begun to acknowledge that it is time they passed the batting.


This week, we need stuff from Texas Instruments, so Lady Aida, what is the thing that we don't have that means we can't make electronics for all the kids out there who really want to build robots and more, what do we need? We need some more of the DRV 8833's and the PWP package, although not too picky, I can always redesign my boards by preferring just to get in this package. We have a bunch on order, I think we have like 8,000 on order, a couple thousand of them will help us. This is a dual H-bridge driver chip from TI. We really like it because it's high current, I think it's about 2 amps and up to 10 volts, so it's great for, you know, we use it on motor driver boards in the Cricket, and we like that package you see with the power pack on the bottom. We can use it with two DC motors, or you can use it with one stepper motor. How nice is that? You want these chips? We want these chips. This is what I want. I want them to rain down on me. It's a rain in chips. Hallelujah, it's a rain in chips. So this is the chip, and what do we use it in Lady Aida? We use, well, we have a break up board, but we use it a lot for that we can't get back in stock is our Cricket boards, which are all in one robotics development platform that you can plug in a feather into, where you can plug in a circuit playground express. We have one for the micro bit, we have one for feather, and we have one for circuit playground boards, the classic Express or Bluetooth, and it's one of those things where it's like we have every other part to make this board except for the DRV8833. And you're wondering like, okay, what type of projects could you do with a Cricket that are now not possible because of this part? And I have a little video that I did from a long time ago. It's so wholesome. And this is a Cricket project that we did here together one night. So I'm going to just play this video. T.I., think of all the things that we could do. I think how wonderful it would be if we could get these chips. So here it is. All right, so how many of these do we need so we could do more projects like that? Look, I need like a reel of like a thousand, couple thousand would be really great. We would use them only for good. No kids would be crushed. Only true love. Love that transcends the human body and the physical instantiation of our spirits. That's right. Yeah. So anyways, I'm going to send this video. To believe in love and send us some DRV8833. I'm going to send this video to Texas Instruments and either we're going to get chips or they're going to say never contact us again. So that's this week's chip shortage.


Hi, I'm Ross Dawson, a futurist together here with Gerd Leonard, my friend and fellow futurist. And having some conversations today about the future of money. And it's a long time since we bartered with shells and so on, the whole growth of currencies, but now that seems to be opening out in a big, big way. So, Gerd, what do you see in the future of money? Well, five years ago when I was still doing work in the music business and Facebook came along, then we were joking, lots of people were joking, saying, you know, I wish I could pay my landlord as a band with Facebook money, with Facebook, right? And this is actually becoming possible now. I mean, Facebook credits will be exactly this. It's starting an alternate currency, in my view, may eventually be more important than the dollar, than the US dollar. I mean, this is a global currency that you can spend on stuff that you can buy on Facebook, including real-life tickets, airplane tickets, cars, and so on. But you can also earn it. This is the important part, right?  This is the thing that bloggers have always dreamed of, getting paid by the audience, right? So that's Facebook credits and we have virtual currencies and we have Cochino and Flatter and so on. So we're looking at a world where it's probably going to be not the coins and bills in our wallet, but that's going to be gone, the credit cards will be on the mobile phone, and there's many ways of paying each other through non-cash, right? This is going to be a very big shift also for the banks, of course. You know, a fascinating space. I mean, certainly I want to get rid of the small change in my wallet and I think it's kind of sad that in 2011 we haven't been able to do that yet, but hopefully it won't be long. Clearly one of the reasons why we're still going to keep some, I think the dollar bills will still remain, or the hundred dollar bills rather, is that people like buying illicit substances, for example, and not doing that in a traceable way. But obviously one of the most interesting things just recently has been Bitcoin and its rise, particularly because it's basically untraceable. And there are some ways in which you can pack it sniff to be able to see some of the transactions, but basically Bitcoin is a completely open way of exchanging financial money or without it being seen by governments. And that's a critical point because governments still want to make control over their currencies. They still regulate alternative currencies and barter systems in order to be able to tax transactions. So if you want to see the rise of these basically cryptographic financial systems, that completely changes the landscape, or might completely change the landscape. I mean if we're looking at the fact that the other three billion in the developing countries, Brazil, Russia, India, China, Indonesia, and so on, they're all coming online now and they're going to use their mobile phones to go online primarily. The mobile phone is the bank and they're called the unbanked for a reason, right? Yes. But there are no banks and many Africans don't have banks, right? And they don't have power to use a computer. So the mobile phone becomes the bank, right? And this is going to be three billion people using mobile phone as banking. And then we're going to be sucked into that as well. So that's a huge shift towards that part of the equation. But also what does money actually mean and who gets paid for what? I mean these kind of things will be completely turned upside down. I think in the future we will pay people not to work, right? Because they can be more beneficial to us as a whole as a society if they're not working in the sense of getting a regular job, right? So it will be paid to pontificate, for example, or to create things or to just think about things like academics do today. Paid by who? Paid by who? Paid by us, by the users, right?  So you'll use social networks to fund group efforts. Well, I suppose that's the micro-payments story and I think it's been interesting where if you look from the late 90s when people were talking micro-payments, strictly micro-payments for content that that was a compelling story. So far we still haven't seen that. I still think the jury's out. It still is a compelling story. I still believe there is the potential for micro-payments yet we still haven't seen the evidence that this is going to really happen. You need huge volume for this, right? Facebook will have a billion users by the end of the year. QQ, which is the biggest Chinese service, already has 700 million. They also have a billion by the end of the year. They will all go public, right? These companies, Tencent and so on. So what will happen then is that these companies can create ways of payment and micro-currencies, right? They'll become global ways of generating this way. So far we haven't had enough juice in the network to say a ban can go and say I need $50,000 for the next record. That just wasn't possible with some exceptions like Kickstarter and so on. But by and large not possible. But when you have 2 billion people connected, right, and you have a fellowship, you can raise money in this way and you can also turn what's called social capital into real money, right? There's a conversion process. Yeah, it's interesting. I suppose there is in fact a continuum between the crowdfunding models of Kickstarter or Indiegogo and so on with micro-payments. I think they're both people contributing to see something happen they want. But on slightly different scales. Obviously Kickstarter or the crowdfunding platforms, you have a smaller number contributing more. The micro-payments, it's hopefully more people contributing less. There still needs to be this habit or habitual way of being able to do that. And this fact, for example, being able to go to a particular website page, you charge two cents to visit that page, and that just becomes part of your everyday. Clearly, and in terms of mindset of getting there.  I mean, developing countries don't have a problem with changing habits, because they're moving like this. But we do, because we're saturated, right? So there's a whole bunch of things that need to happen for that first. But I think in general, if you look at what's the very definition of money and payment and currency will substantially change the next five to ten years as a consequence of technology. I mean, the fact that I can be tracked by transactions is a big issue already. But the way of cloaking that or safeguarding that is being invented right now, because it's needed for everything, not just for money. And of course, money, the other thing is money is content, right? And all content is moving into the cloud. So films, television, and money is moving into the cloud. So that's inevitable. So if you're in the banking business or insurance or loans or whatever, I think that you have to face reality that money is going to be in the cloud as well. Well, thanks for tuning in. You can visit Ross's site. What's your site? RossDawson.com. Or just search for Ross Dawson. Okay. And mine is mediafuturist.com. And thanks for tuning in and listening to us delivering our diatribes. Thanks.


Hello again, we will be going to the next lecture today. We will be discussing on discrete probability distributions. The reference books for this topic are presented here. Montgomery and Runger's book Applied Statistics and Probability for Engineers, 5th edition, Wiley India 2011 and then you have the book by Organiak random phenomena okay by CRC Press. In the previous lecture we saw about random variables okay. When we conduct a random experiment we are unsure about its outcome and so the different possible values are called as random variables and they may take some specific real values okay. The possible values of random numbers may be discrete entities or they may be continuous range of numbers. What I mean by continuous range of numbers is it may be falling between a lower value and upper value okay or a lower limit and an upper limit if you want it like that. But in between these two limits okay it can take any possible value. As an example if the random variable mole fraction is considered from an experiment okay a distillation experiment for example. The mole fraction can take values only between 0 and 1 okay both included but in between 0 and 1 it can take any value 0.12, 0.123, 0.1234 whatever it depends upon the accuracy of your measurement device okay. Of course the calculator can give any value of the mole fraction running up to 7 or 8 digits okay.  In this case the random values or the random number values are discrete okay. So we are going to talk about a very interesting function the probability mass function okay. The probability mass function f of x assigns a probability value to each of the discrete values of x there is a typo here which I am correcting okay. So earlier it was capital X here f of capital X now it has been changed to f of x. If you recollect the random variable function has converted the original sample space into real numbers and these numbers are assigned probability values. This we covered in one of the previous lectures okay. So the random variable may take discrete values x1, x2 so on to xn okay. So we now define a probability mass function and denote it as f of xi okay and the role of the probability mass function is to assign a probability value to each of the possible values xi okay. So please note the definition again probability of x is equal to xi that means what is the probability that the random variable capital X takes the value small xi and that is given by the probability mass function f of xi okay. When you write down the statements pertaining to statistical calculations you have to make sure that the terminology you use is correct okay the notation you use is correct. You cannot write f of xi okay. You do not have capital xi you have only small xi you have capital X and small xi but you do not have capital xi okay because once a random variable is assigned a value it becomes small x of i and as we stated earlier f of xi should be greater than or equal to 0 for all the values of xi all the possible values of xi. The means that the probability values are all positive. Now the other important thing to observe about the probability mass function f of xi is the sigma i equals 1 to n f of xi should be equal to 1. It means that the sum of all the probabilities is equal to 1. You have different entities each entity is having a probability value and the probability values are assigned by the probability mass function in such a way that when you add the total probabilities they should be equal to 1 okay. The probability value assigned to each of the xi may be same or they may be different okay.  Similarly the probability of a number 2 coming is also 1 by 6. So you have 6 numbers and all of them have share the same probability of 1 by 6 okay. On the other hand there may be other examples where you may have a discrete number of possible outcomes but all of them need not have the same probability value okay. They may be having different fractional probability values but when you add them up they should become equal to 1 that is very important. Mu is equal to e of x okay that is the mathematical notation here. Mu means the mean of the probability distribution okay. e of x means the expected value of the random variable x okay. The expected value of the random variable x is defined as mu is equal to sigma i equals 1 to n xi f of xi okay. The mean is also referred as the average okay and when you take any cricketer okay and he has gone unperformed in a test match series or even when he is coming out to bat after the previous wicket had fallen okay. The first thing which is shown on TV is the number of matches he has played, the number of runs he has scored and then the average okay. The average is a measure of his overall performance okay. If a batsman has scored an average of 50 then he is supposed to be very good but it does not mean that he will score 50 in the current innings. He may score 100 or he may score a duck okay. So the mean refers to the center of the probability distribution okay. Imagine you have a long scale and on the scale you have markings corresponding to the possible values of the random variables okay and each of these values will have a probability assigned to it okay. Let us see that you have numbers 1, 2 and 3 and each of these numbers 1, 2 and 3 will have a probability value assigned to it okay. Number 1 may have a probability of 0.2, number 2 may have a probability of 0.3 and number 3 may have a probability of 0.5 okay. So you put coins on the scale or you put weights on the scale adjust on these numbers. You put weights and you put the weights in such a way that it is proportional to the probability value okay. So if you put coins for number 2 you put more coins for number 3 and you put even more coins for number 5. The number of coins or weights you put for these values will be in proportion okay and then you put the scale on a knife edge and see where the knife edge will balance okay and the knife edge will balance at the mean value. Now let us come to the variance. We know from our earlier discussion because of the variability in the data you had to find the average. The batsman was not scoring 50 in all the innings okay. He was scoring 0 in 100 in the next 30 in the third 70 in the fourth. So when you take an average over a long period of time you get the average of 50. So the performance of the batsman is variable okay and that is why you had to find the mean or average value. Now the variability is quantified in terms of another parameter called as variance. We refer to it as variance of the random variable x and that is defined as the expected value of the deviation of the random variable from the mean squared okay. So what you do is you first find the deviation of the random variable from the mean mu and then square it. This is called as deviation squared and the expected value of this deviation squared or the expected value of the square of the deviation is equal to sigma squared okay. So now by using the definition for expectation we write v of x is equal to sigma is equal to 1 to n xi minus mu whole squared f of xi okay. Please do not forget to put f of xi okay. You may recollect that the sum of all the deviations from the mean will be equal to 0. When you square the deviations from the mean you take the sum they will not be equal to 0 okay and each of such squared deviations okay should be multiplied by the corresponding f of xi. When you expand this sigma is equal to 1 you get xi squared f of xi minus mu squared. This derivation is quite simple and you may want to try it out yourself. The variance is a useful measure as it indicates the spread of the probability distribution okay. Even if there are 2 distributions having the same mean or average it does not imply that they should have the same variances. You may have the same average but you may have different variances. For example in one class the average may be 15 another class the average may be 50 okay. But in the first class the lowest mark may have been 20 and the highest mark may have been 100 okay. So average is still 50 when you add up all the marks and divide by the total number. The second class the lowest mark may have been 40 and the highest mark may have been 60 and the average would have been 50 okay. There is lot more variation in the first class in the marks when compared to the marks in the second class even though both of them had the same average. By definition the square root of the variance is the standard deviation. Variance is denoted by sigma squared and so standard deviation is denoted by sigma. As I said earlier we should look at the notation or terminology issues whenever we use the language of statistics okay. The grammar has to be correct. So let us look at some of the notation issues in the usage of the probability mass function f of xi okay. So can we denote the probability mass function as f of x instead of f of xi okay. You cannot put capital X in the argument of the probability distribution function or the probability mass function okay. It should be f of small xi. So writing the probability mass distribution as f of capital X is equal to capital X plus 3 by 25 makes no sense okay. If the possible random sample space values are 0, 1, 2, 3 and 4 then to find the probability of the occurrence of one of these values xi we use the notation f of xi and not f of capital Xi okay. The following form which I have shown here is correct f of the random variable X taking a value xi is given by f of xi and that is equal to xi plus 3 by 25 okay. This is correct notation. The function f of xi becomes alive and takes a value only when the random experiment has been performed and X random variable capital X can take a particular value xi okay. Before throwing the dice or before throwing a die the random variable was X.  Then you can find what is the probability of this number xi equals 1 or 2 occurring and you can estimate that value. Sometimes you may not even know the probability beforehand okay. In the case of a die you know that all numbers are equally probable and so you can beforehand itself say that probability of a number 1 occurring is equal to 1 by 6. But there are many random experiments whose outcome is not known for sure and you also cannot predict what that probability of that outcome is going to be okay. So you have to conduct the random experiment obtain the value of xi and then find out the value of f of xi the probability value okay. Generally we can represent the probability density function sorry it is not probability density function. Probability density function is used for continuous distributions. Now we are dealing only with discrete distribution. So the probability distribution function or the probability mass function f of x is given by x plus 3 by 25 small x is used throughout. And this function is evaluated at each possible value of xi okay. So we saw about the mean when we write the definition for mean mu we write it as expected value of capital X okay. Expected value of the discrete random variable X capital X. Mu is equal to e of x but it is not equal to xi f of xi where the xi's are capital xi this is wrong. When we actually go about calculating the expected value then we use small x and f of xi and the argument in f of xi is also small xi okay. So do not use capital xi in f of xi use xi into f of small xi when finding the value of mean. So this is important notation or terminology. Similarly when you are defining the variance of the random variable X it is expected value of e of x minus mu whole squared is equal to sigma squared but this should not be written as sigma is equal to 1 x minus mu whole squared f of x okay. You should not use capital X here but you should rather use small xi. We will take a simple example okay. So let there be a lot of n balls of which B or blue in colour and the remaining are red okay. I am sure that this kind of problems or variance of this problem you would have seen several times in the past. So let us define the random variable as the number of blue balls that have been picked. The random experiment is performed by taking two picks from the lot okay. You take one ball, note the colour, put it back, take the next ball, note the colour and then put it back okay. So this is called as picking and replacing. Let P denote the probability of picking up a blue ball and Q the probability of picking up the red ball okay and the question is you have to define the original sample space, the random variable space, the probability mass function and verify whether its properties are satisfied okay. To summarize P is the probability of picking a blue ball and Q is the probability of picking the red ball. So we have to find the original sample space, random variable space, probability mass function and see whether the properties are satisfied. The original sample space is given by BB, BR, RB, RR okay. So the first event is both of them are blue balls. The next event may be blue ball and red ball or red ball and blue ball.  So since the random variable X was defined as picking up the number of blue balls okay. So the random variable X is denoting the number of blue balls which have been picked from the random experiment. You have the possibility of 0 blue ball or 1 blue ball okay either of these two or 2 blue balls okay. So these are the possible outcomes expressed in the form of the random variables okay which take the values of 0, 1 and 2 okay. The possible outcomes are no blue ball, 1 blue ball or 2 blue balls. So the original sample space which was having 4 entities was reduced to only 3 entities in the random variable space. So the probability of the random variable taking 0 okay that means no blue balls have been picked.  Similarly f of X is equal to 1. The probability is the first ball picked was blue, the second ball picked was red. So the probability would be pq or the first ball picked was red and the second ball picked was blue.  Now you can verify whether the sum of the probabilities is equal to 1 okay.  So sigma f of X i should be equal to 1.  Since the probability of picking up a blue ball is 1 minus probability of picking up the red ball okay or the probability of picking up a red ball is 1 minus probability of picking up a blue ball. We have q is equal to 1 minus p, you put q is equal to 1 minus p here you get 1. So you are having the sum of the probabilities totaling or adding up to 1 okay. Obviously the probability values are fractional. You have 8 balls in a box, 3 of them are blue and 5 of them are red. Then the probability of picking up a blue ball would be 3 by 8 and the probability of picking up a red ball would be 5 by 8 okay. So you can see that q is equal to 1 minus p. Next we come to another interesting and important function, the cumulative distribution function capital f of X and that is given by f of X is equal to probability of X less than or equal to X okay. What is the probability that the random variable takes any value which is less than or equal to X okay. So that is the cumulative distribution function and that is written as sigma X i less than or equal to X up to n f of X i okay. So you are summing over all the values of X i until you either go below X or reach X. There is an interesting distinction between the probability mass function and the cumulative distribution function okay. In the probability mass function you can only apply the function to the possible values of the random variable X okay. So you can only apply the probability mass function to the X i values. If for example you have another value Y i which is not equal to any of the possible X i values f of Y i is equal to 0 okay. But when you do the cumulative distribution function this X value need not be X i all the time. It may even be a value between 2 X i values between 2 permitted X i values. For example if the permitted X i values are 0, 1, 2, 3, 4 and 5 you can even find the cumulative distribution function of random variable X less than or equal to 3.5. So that is permitted and that value need not be equal to 0 okay. So you are adding up the probabilities as we saw in the mathematical equation. This is the cumulative distribution function. Cumulative means adding or totaling okay. So that is why you add up the probabilities up to the value of X okay. That X may be equal to allowed value of X i or it may be not equal to X i okay. And the cumulative distribution function ranges between 0 to 1 okay even though you are adding the probabilities and the maximum value would be equal to 1. Because the sum of the probabilities is equal to 1 and if X is less than or equal to Y f of X is less than or equal to f of Y. So you can either write it as f of X or f of X i okay. But you cannot write f of capital X. Whenever you are doing the cumulative distribution function calculations there is a possibility of committing mistake okay. So let us see the probability mass function is given to you and you have to find the cumulative distribution function okay. And the statement may be similar but slightly different. Small differences can be noticed. In one case you have to find probability of B less than or equal to X less than or equal to E. In the second case it is probability of B less than X less than E. Third case probability of B less than or equal to X less than E. In the fourth case it is probability of B less than X less than or equal to E okay. So you may think that all of these would result in the same value okay. But that is not going to be the case as I will demonstrate in a, with an example. Can we write probability of B less than or equal to X less than or equal to E as f of E-f of B. No it cannot be written in this fashion. Let us take an example okay. So you are having a cumulative distribution function f of x, small x mind you and these are the possible outcomes A, B, C, D, E, F. These are the possible values taken by the random variable and these denote xi values, small xi values. The probability of A occurring is 0.1, B occurring is 0.2, C occurring is 0.3, D is 0.2, E is 0.1 and F is 0.1 okay. These are the probability values okay and the length of this line corresponds to the probability. This line is twice as long as this line because the probability is 2 times higher okay. Now let us see the probability of the random variable X taking a value less than or equal to E less than and greater than or equal to B. So this is what you want to find out okay. You want to find the probability that the random variable will take any value between B and E both included, both B and E included. So the probability of the random variable taking the value small b is 0.2, C is 0.3, D is 0.2, E is 0.1.  So that is the value you get okay and that may be written as probability of X less than or equal to E minus probability of X less than B. Probability of X less than or equal to E is 0.9, 0.1 plus 0.2, 0.3, 0.3 plus 0.3, 0.6, 0.6 plus 0.2, 0.8, 0.8 plus 0.1, 0.9 that is why you get 0.9 here and then you get probability of X less than B, less than B is only A and the probability is 0.1, it is 0.9 minus 0.1. The second case is probability of B less than or equal to X less than E okay and that is equal to 0.7. Here B is included but E is not included okay. So it should be less than E. So we count 0.2 corresponding to the occurrence for B, probability of occurrence for B. So 0.2 plus 0.3, 0.5, 0.5 plus 0.2, 0.7 okay. We are excluding E because it is probability of X less than E minus probability of X less than B okay. Probability of X less than E is 0.7, 0.3 is 0.8 sorry 0.1 plus 0.2, 0.3, 0.3 plus 0.3, 0.6, 0.6 plus 0.2, 0.8 minus probability of X less than B okay. Probability of X less than B as we saw in the previous case is 0.1. So you get 0.8 minus 0.1 which is 0.7. Similarly you can easily show that probability of B less than X less than or equal to E is 0.6 and probability of B less than X less than E is equal to 0.5 okay. So coming back to our original problem statement what is the probability of B less than or equal to X less than or equal to E that was not equal to F of E minus F of B but it is probability of X less than or equal to E minus probability of X less than B okay and that was 0.9 minus 0.1 which was equal to 0.8 okay. So do not try out some formulae based on intuition okay in these kind of situations okay rather than using these formulae I would rather advise you to construct the distribution of probabilities and then do the calculations. Anyway the mathematical formulae are here but from my point of view these formulae need not be memorized but they can be easily implemented okay by just using the diagram. Now we are going to look at something known as moment okay. In physics you might have seen the term moment coming to use rather frequently okay force into distance okay. So here also we have ordinary moments and central moments. So I will just give the definitions here. So the expected value of E of X is given by i is equal to 1 to n X i F of X i this is called as the first ordinary moment. In general expected value of G of X is equal to sigma i is equal to 1 to n G of X i into F of X i okay. Here you had random variable X so you put X i here. Here you have G of X and so you put G of X i here. The function defined by the random variable is implemented on X i inside the summation okay. So if we define G of X as X to the power of k then the expectation of G of X is called as the kth ordinary moment of the random variable X. So the kth ordinary moment is represented by m of k is equal to expected value of X of k. The first ordinary moment is represented by m1 which is equal to mean and that is equal to expected value of X because k is 1. So you have expected value of X and that is equal to mu. So far we have been defining the function G of X as X power k but we can also define the random variable X as a deviation from A where A is a constant value okay. Earlier we had by default used A as 0 but it need not be the case always. So we want to put A as a constant value and we are going to find the kth moment okay and remember k should be an integer value. So G of X is equal to X-A to the power of k. The expectation of G of X in such a case is called as the kth moment of the random variable X about A okay. The kth moment of the random variable X about A. The moments about the mean are defined as mu of k is equal to expected value of X-mu to the power of k. So k can be 0, k can be 1, 2 and so on. Remember k should be an integer okay. So here we are finding the moment about the mean okay. The mean is the center point of the distribution okay. So it is X-mu, expected value of X-mu to the power of k is called as mu k and these are referred to as the moments about the mean for different values of k. It is also termed as the kth central moment of the random variable X. Since we are taking the moment about the mean, we refer to it as the kth central moment of the random variable X. So when you put k is equal to 0, mu 0 is E of X-mu to the power of 0, expected value of 1 is 1. Mu 1 where we put k is equal to 1, so expected value of E of X-mu to the power of 1 is expected value of X-mu is equal to mu-mu is equal to 0 okay. So this is the expected value of X-mu okay that we call as mu 1 which happens to be 0. So mu 0 is 1 and mu 1 is 0 okay. The second central moment is termed as the variance. The central moment is taken with respect to the mean. So k is equal to 2, the expected value of X-mu whole squared is equal to sigma squared. This definition we had seen previously but now this is called as the second central moment or the second moment about the mean. The square root of these variance is called as the standard deviation. So E of X-mu whole squared or mu 2 may be written as m2-mu squared, second moment-mu squared okay, second ordinary moment-mu squared. So you have found the mean variance and the standard deviation. So you have to scale it properly okay and so you want to see what percentage of the mean or what fraction of the mean is the standard deviation and so you write the coefficient of variation Cv as sigma by mu. It provides a dimensionless measure of the relative amount of variability exhibited by the random variable okay. If sigma is very high okay, sigma is 100 okay. You may think that there is a lot of variation, the standard deviation is pretty high but you have to find it with respect to the mean value. If the mean is in the order of 200 okay and the standard deviation is 100, 100 by 200 is pretty high okay but if the mean value mu is in the order of 100 by 10000 okay which is 1 by 100 and that would be 0.01 which is pretty low okay. So you have to compare the standard deviation with respect to the mean value. The third central moment or the moment about the mean mu is defined as skewness okay and that is given by mu3 is equal to expected value of x-mu whole cube okay and the significance of the skewness it illustrates or indicates the asymmetry of the distribution okay. You can understand that when there are several values of x, they may be distributed about a mean value okay and they may not be distributed uniformly with respect to the mean value and so you may have negative deviations and positive deviations and the relative difference between the negative and positive deviations with respect to the mean is given by the third central moment. So third central moment or skewness is a measure of the deviation from the symmetry okay and the dimensionless quantity mu3 divided by sigma cube is known as the coefficient of skewness. If the distribution is perfectly symmetric both these skewness and coefficient of skewness both vanish when the distributions are such that the negative deviations are more than the positive deviations from the mean, the distribution is said to be skewed to the left okay. There is dominance of negative deviations when compared to the positive deviations and so the distribution is said to be skewed to the left and both mu3 and gamma3 take negative values. The fourth central moment is called as the kurtosis and that is defined as mu4 is equal to expected value of x-mu to the power of 4 okay. It is a measure of the flatness or the sharpness of the probability distribution okay. The ratio of the kurtosis to the fourth power of standard deviation is termed as the coefficient of kurtosis okay. A distribution with the high value of kurtosis has a sharp peak. On the other hand the distribution with low value of kurtosis is flat or rounded. When the gamma4 value is less than 3 the distribution is said to be mildly peaked or platycurtic okay. When gamma4 is greater than 3 the distribution is said to be sharply peaked or leptocurtic. These terms you may encounter in standard texts or even in research papers showing distributions and explaining their characteristic features. You may often come across these terms platycurtic and leptocurtic. So depending upon whether the peaks are sharp or the distribution is flat and rounded the appropriate terminology is used. The next topic in our discussion is median for discrete probability distributions. The median xm for a discrete probability distribution is the point within the range of the allowed values of the random variable x such that the cumulative distribution value at xm is exactly 0.5. In mathematical form sigma i equals 1 to m f of xi is equal to the cumulative distribution value at xm should be equal to 0.5. F of xi is the probability mass function. The xm value here is such that the probability of x less than xm is equal to probability of the random variable x greater than xm are both equal to 0.5. This concludes our brief discussion on the discrete probability distribution functions. We will be now moving on to continuous probability distributions. They are also called as the probability density functions.


It's really abrupt, you know, it's about ten times faster than anything in the geological record that we know about. As the status of the ocean certification increases, we're going to start to see effects on organisms that people rely on for food. In the last ten years, there's been a huge increase in our ability to communicate to the general public that the ocean ecosystem isn't just something that sits there and is stable.


Hello, and welcome back here to the channel. I'm EvilRabbit We're here on Drift21 to do some challenges in the BRZ to unlock some more money as well as try and unlock some more Treks, so make sure you guys follow me and you tap that subscribe button if you're not already a subscriber I would appreciate the support definitely helps me grow the community as well as tap that like button Definitely helps me out as well. So without further ado, let's get to it So hopefully we can use the BRZ in this challenge. We are just going to see if we can BRZ it's gonna let me use the BRZ. Okay So we're gonna rock the BRZ in this challenge try and just complete it a little more speed in this car This is the one we did in the Supra or it's a Supra in a 350 We're we able to get a run we failed the first clipping Challenge fail. So we're gonna retry that We missed the first clip apparently. We did get that one So here you can in this one Got that one too Then we got to get this front inner clip We should have been able to complete that. Hopefully we got enough points We didn't get a trophy so we're gonna move on to the next challenge because we did complete that So we're gonna move to the next challenge a very similar challenge drift zones and things of that, but We're gonna see how many we can get done with the BRZ in this form and then we can add some more power to it That backfire though. We actually got that one. Oh We're way off that one. Oh boy. We didn't miss that last one We definitely failed. Well, yeah, we failed that last coming down. We did get a trophy We didn't gain some credit see but we're gonna definitely read Try that see if we just can't get all the drips out We got to get some money involved so we can actually upgrade the turbo in this car It's more power on the BRZ Okay, we totally missed that and we missed that one So we got that one, but we missed the first two Yep, nope, definitely gonna retry that. We start third then second gear all the way Definitely using that wise fab Part shop max for everything got in this game. I think we got them all that time I'll be happy if we completed them all we did we got a silver trophy so more money on the docks So what next challenge we go next challenge a little bit tighter, which might be better for this BRZ being lower power We're gonna see what we can do with it on this next challenge One of the challenges in this car sounds super We miss that one and that one. Where's my restart button? There it is This may be hard to actually get Given the current configuration as BRZ And we may have gotten that one too It's part of the second gear hero right now. We may have actually yeah, and we got a silver trophy I'll take it Silver trophy next challenge. We're just banging through these challenges right now with this one there It looks like there's some speed involved. So probably third gears is that big section See what you do I'll take it that was actually impressive. We did get a bronze though. So that's good Next challenge Nice and tight little section. So second gear probably hero for this BRZ. We're just banging through these challenges right now I'm not sure if we actually accomplished that. I think we may have We did but we did not get enough to complete it So we definitely gonna retry that All the bangs and pops. I think that's where we lose all our points And I restart that one. All the clutch kicks. So it's such a better run First person action. See what we do it first. Now we're near that clip. We try that again About one. I cut that a little bit too far in. I'm unable to get it that time.  I Don't think we got enough We did. We did get a silver trophy. So I would definitely take that move on to the next challenge We're banging these out here with this BRZ. Here are Jeff Tuneau If you guys want to keep seeing more Jeff Tuneau make sure you guys follow me on all social media. I'll check out the description box below. Oh boy Already cutting it in. Oh, how we actually okay, so we missed that one. Yeah, this And this one's a much Tighter course, okay much tighter course. Oh did I fail that? I'm calling a rerun on that one because I'm not sure how I failed that If you guys know how I failed that let me know down in the comments because I'm not sure so we didn't fail that one Okay, so that's how we're gonna get that one. Okay. All right. We got this now more speed those in this section Now we didn't count either It's not fun when we missed the one that we've been getting and are able to get the final one, but you know We'll get it. It's gonna start this drifting all the way. All right. We got it now We definitely got that challenge. I'd like with us with a bronze trophy Next challenge this one looks like it's a time attack. So now gripping just speed This would have been better if the motor wasn't turboed because we had a lot more grip, but we're just gonna go full set Third gear to try to keep the car from spinning Oh understeer major understeer But we got a silver trophy Not too bad I feel like I shouldn't have broke on that section. So we're gonna try it again Because I know where we're going now. It's a hard turn Much faster gold should be there we go gold trophy On to the next one This one's gonna be a little more complicated because it looks like there's a small little s section in there But we got this we knew pretty good in the brz. So If we were in the right gear, we'd be okay. I don't know how that worked. I don't know if that was good enough 21 seconds out of a gold. Yeah, no, we gotta get this gold I see where I failed it to pull the handbrake. Oh, there's too many grippier tires No, it's not gonna. Oh, we were even slower Well Give it one more final go with this one. It needs the body of the car here much as possible Come out this way off throttle down shifts. There we go. Bring it out Bring it in don't rest here as much kind of apex in that. Oh, we got it. Oh, we got it That's what I want to see another gold So the next challenge is another time attack We're getting plenty of money to upgrade this brz and uh, we're completing challenges one after another Definitely locked up the brakes there Trying to manage the the slip I don't think we got it. We got a silver though. We were 16 seconds We'll give it one more try for gold, but this one We gotta try and manage the slip with grip, but we do not have grippy tires on this car. So It's definitely where We're hindering it. Definitely lock up the brakes is not helping Um, not sure. I think we got it. Um, no point 79 Unfortunately, we did not get that one I feel like we do need to upgrade the tires to something grippier if we want to complete these time attacks because this is uh A little bit challenging now I think you can see really amounts of understeer Yeah That looks gonna be tough to get. I mean silver is okay Look at all the gold and money that we are getting with that I think this might be one of the last few challenges unless we just go full on I'm sure we unlocked a new track for sure If you guys who want to keep seeing this, let me know down in the comment section down below Oh, that's not gonna help us when we hit that It's it's like trying to autocross in the rain But we got a silver bite Very close. We wouldn't hit that code. We may have gotten a gold That lock up of the brakes not gonna help me out in this section. Nope. We're not gonna get it that time So let's go to the next challenge. This looks like a full time attack This might be the last one of this map Oh, man, the brz needs some front grip. It's definitely uh Need some some grippier boys in the front. There's an understeer. I don't know where I'm going I don't know the track. Well, it was kind of a shift lock. I mean I don't know where we completed in that one. That's a silver You know for not knowing where I was going I'll take that silver next challenge. What is this? Oh Jim conna in the brz. Okay. I think we're gonna end this on the conna We'll take a look what kind of money we got for it to build for in the next episode See we got you got two minute Jim conna. So it's kind of a freestyle runs Oh Kissing the wall Another wall tap That's a major. Let's see if we can't do a little uh We're in any roundabout 240 donut Definitely very impressed with how this brz is driving with just this middle two. We're gonna try to get this again We're gonna jump it this way Oh major major understeer.  Okay. They only give us max 360. Okay So I'm not sure what we got Oh, we got silver trophy though I'll take it way off of gold definitely needs some more power for that So let's go back to the garage And see where we're at money-wise because I'm sure we got plenty of money to boost up this car Maybe put some better suspension on it some wider grippier tires Because this brz could benefits from some grippier tires And maybe some upgraded suspension because we do have angle kitten all there which we need to adjust So how many challenges do we have left in this one? So we only have one other challenge left in this and it's another gymkhana. So we're gonna say that for another episode Oh, we got 17 grand We have 17 grand. So this car is going to get some uh some bigger tires And maybe some different wheels. We got to throw some different wheels on there probably some five spokes And we're probably gonna space them out and throw some grippier tires on us We do have a lot of grippy tires, but we're gonna do some other grippy tires So make sure you guys follow me on all social media all the time in the description box below And as always I thank you guys for coming back and watching. I'm evil rabbit. I'll see you guys on the track


I'm going to preface this review by saying I know absolutely nothing about comics. I'm not a huge fan of the genre. I do enjoy it when it is very, very well done, but I'm very picky. So keep that in mind as we go into this, that I am not a comic book aficionado. I don't claim to be. I know very little about the process. I know very little about the characters in this book other than what I have seen in the movies. My full, my knowledge of Batman extends only to his theatrical universe. On with the review. Hello, everybody. E here. Welcome back to another comic book review. I've done a couple of these in the past, only when something really stands out that is really, really good or really, really bad, do I bother doing a review of a comic book series. Or when somebody sends me something for review. My friend, Lertis, he has sent me stuff. I think someone else sent me something that I never actually got to because it wasn't my deal. And just flipping through it, I was like, this is pretty bad. So if you're that person, I apologize, but I never read it. Just being honest, that's me. But today we are talking about Batman hush. The whole reason I read this, first off I got comicsology because I have so little time nowadays. I want to try and still continue reading, but and I can consume audio books and comic books about the same speed. So I'm blowing through a lot of audio books and I'm going to be going through a lot of comics, maybe two a month for you guys. I'll probably be reading more, but I won't be reviewing them all because once again, I don't know that much about comics. Maybe one day I'll get better. But what we're talking about here is the reason why I picked this one up. So there's all these rumors that the new Batman movie with Robert Pattinson, Colin Farrell, Paul Dano, numerous people, numerous big name stars are going to be in the next Batman reboot called The Batman, I think. And people are speculating that this is the hush storyline because we've seen a penguin, we've seen, I don't want to spoil something else. We've seen something else that kind of spoiled it for me. People talking about it online. So if you don't know the storyline, stay away from it. You'll probably enjoy it more than I did. And I did enjoy it. I did enjoy it. Don't get me wrong. I liked it, but there are some valid complaints that I want to bring up that people are having. I both love and disliked how many characters there were in this. You got Batman, Superman, Catwoman, Huntress. Then you have Joker, Riddler, Penguin is even in a couple of frames. There's, let's see, a hush is a new villain. So many that like Killer Croc, Clayface, all these people are in this huge rogue gallery of villains. Scarecrow is in it. And I hope these aren't spoilers, but I think they're all on the cover. I'm not sure. Harley Quinn's in there. But how they all tie together is what I liked. I I'm not a huge fan of piling in this many villains. And I understood why it fit the tone and the themes of the book. I get it. They're going back to Batman's past. I understand it. But here's the thing. It just went on and on and on. There there are several sections of this comic where they're having a really cool fight and we get stuck with Batman's internal monologuing. And I think it takes away the thrill of the fights. That's my opinion. How else are you going to get this information? That's so important. They need to put it in the thoughts while he's fighting. I don't know if they could have done it better. Once again, I don't read too many comics. So is there a better way they could have done it? I don't know. What I'm saying is I enjoyed the comic book for what it was. And I do hope that this is the plot of the movie, especially because of the twist. Unfortunately, the twist was actually ruined for me because everybody I saw talking about the movie was talking about a certain character and his character arc and that literally spoils the twist of who Hush is. So please try and stay away from spoilers, because if the movie is the same as this comic series, you were I would say away from the movie. Anyway, I know Batman fans and comic book fans are mad at that already. But I would say wait for the movie because there was one. They force a red herring really, really hard. And as long as you don't know who the character is, it's fine. But if you don't, this comic book is kind of on the boring side. I did like the set pieces. I did like the action. I did like the adventure. But there were plenty of times when I was taken out of the story by the authors trying to put me deeper into the story. I know this is this sounds like I'm crapping all over this this graphic novel. I'm not. I enjoyed it. I read about two, basically two issues a day. And, like I said, I did enjoy it. But fans, I also went back and read fan reviews. And there's some very valid complaints about how many characters there are and how when you have that many characters, it takes it takes the I don't know, it takes the spotlight off of the story and it doesn't give those characters time to shine. I feel and this is another thing that I got from someone else's review. There is someone else said that. There's not enough that you shouldn't start here, which is what I did. Unfortunately, I should have read the reviews first, but I try to stay away from things like that. I'd already been spoiled. I didn't even know it. I'd already been spoiled on Twitter. But you have to know the history. You have to know they try to build up the history. But you have to know the history of Batman, Jason Todd, Dick Grayson, whatever the other one, Tom, whatever. I don't I don't know. But it's all all the Robbins, all three Robbins. You have to know that history. You have to know all the history of all these roads. The road house of whatever roads, galleries, what's called all these villains. You have to know all their backstory and everything. And they try really, really hard to give you like a little little explanation and description of where these characters come from. But they do it at the most inopportune time. It's right when they're starting to fight or right in the middle of the action. And I'm like, I just want to get to the action because we all we all know who Joker is. But I don't know that I know who Scarecrow is. I don't know that I know who Clayface is. You know, I don't and Zazz, I don't know. He's not in this. I don't believe. But Zazz is a character from the video game that, you know, I don't know his backstory. So I don't know if this is like a running theme in comics.  I don't know. But I think my final thoughts here again is I enjoyed it. I had fun with it. But once I backed away from it and gave it a more critical eye and read some reviews, I was like, yeah, this really does have issues. But I'm also not the type of person to say that I'm going to change my opinion on whether or not it's good or not based on other people's opinions. Because while I was reading it, I was enjoying it. I was having fun. Does it have problems? Sure, it does. Did I notice while I was reading? No, I didn't. So if you if you want a more realistic review of I would say go over to Goodreads. Be very careful of spoilers. I would say go over to Goodreads and read some of the critical reviews, some of the three star reviews. If you want a better critical review for me, I kind of loved it. But once I step back from my name, maybe that was but I enjoyed it while I was reading and that is all that matters. So have you read Batman, Hush? Let me know down in the doobly-doo whether or not you loved it, whether or not you hated it. If you loved it or hated it, tell me why you loved it or hated it so that we can have a discussion. But until next time, I have any you have in you. This has been another comic book review. I'll talk to you guys later. Bye bye.


Hey folks, have you heard about Developer Days? It's our identity conference where you can learn about how to take control of identity, the power of SDKs, and so much more through internal and external speakers. And the best part is, well, Corey Welles and myself are actually hosting life from the New York office. So maybe that's not actually the best part. The best part, wait one second. Did you see last year that we were wearing these super cool Pac-Man jackets? I think that was the best part. No, I'm just kidding. Come learn about all things identity, security, and privacy with us at Developer Days, either virtually at one of our watch parties, or in the New York office. So go ahead and register the night.