 Hi everyone, it's MJ, the fellow-actry and in this video I want to talk about the boundary copulas And essentially this may be start off a little bit with the history So we've already introduced this guy called Maurice René Frichet. He was French He was working with joint distributions in 1956 he kind of teams up with the American Abe Sklar and they publish Sklar's theorem The problem was is they thought that they had found this whole thing up on their own Where it turns out that there was somebody else who 16 years later back in the 1940s had almost beaten them to it and this was this guy called Wesley Hoffden, it's a Finnish name. So my pronunciation is probably not the best Anyway, he published quite a lot of this work in an obscure German Journal and because it was during World War two No one really kind of picked up on it until like we see almost 20 years later. Also Hoffden was working on He was working from negative a half to a half where Frichet was working on more zero to one Which allows more for probabilities So it's amazing how this Hoffden guy was very very close to discovering copulas and their use in probabilities But like I say he scale is just a little bit off Anyway, it's a fascinating history and I do encourage you guys to read up more about it But now in this video I want to talk about their three boundary copulas also known as the simplest copulas and the one that we've already come across specifically in the production video was the independent copula and this basically says if two probabilities have absolutely Zero dependency, thus they are independent We simply multiply the two together in order to create the joint Distribution and this is something like I said We should be very very familiar with from learning statistics and first and second yeah If you're watching this course and you haven't done first and second. Yeah statistics I highly recommend you do that first. Otherwise the statistics here is going to get a little bit more intense Then we have the second copula that I want to talk about something known as the minimum copula and this idea is that The the copula function between f of x and f of y is simply going to be the smaller of the two the minimum value and We also have something known as the maximum copula which says, you know the joint distribution between two marginal distributions is sometimes the maximum of Of the two so Minimum and maximum are probably a little bit harder to comprehend than the independent copula So we are going to talk a little bit more about them, but essentially They all relate to dependency. So when we have zero dependency, we have the independent copula when we have full positive Dependency we have the minimum copula so you can think about this as Sense and dollars if you've got five dollars You've got five hundred cents if you have six dollars. You've got six hundred cents. The two are Fully positively dependent on each other You're just looking at them in different ways that way what we're gonna do if you want to say what's the probability that I have More than five dollars and more than 40 cents. It's simply going to be the minimum of the two We then have something called the maximum which is the full negative So this is when when the one thing happens the other thing does not happen So if you flip a coin if heads happens tails can cannot happen So it's fully negative on on that extent, but let's maybe explore this a little bit more Like I say the maximum the minimum copula are our boundaries. They are our extreme cases So let's look at the minimum one in more detail Okay, if x is equal to this, you know function of y let's say a y plus b and A and B are constants then we can say that x is a monatomic Transformation of y thus x and y have full Dependency and the minimum copula is simply going to be a lower bound Which means if f of x is equal to zero point three and f of y is equal to zero point four Then we're going to see that their joint distribution is zero point three and why this is important It means no matter what two probabilities we have so if we've got let's say 60 percent and 70 percent The whole idea is that we have this lower bound where it cannot be less than this amount And if we know how to come to say the the maximum copula We have this idea where the probability of x occurring is equal to one minus the probability of y occurring and Coming back to some statistical jargon. This means that x and y are mutually Exclusive thus x and y have a full negative dependency You can think of x as being the number on a dice and y is equal to six minus that number on that dice You'll see they the two will be very almost in contrast as x increases y is decreasing so This is our maximum copula and it forms an upper bound It only exists in the bivariate case because as soon as you have more than two probabilities Then things start getting a little bit awkward with something being fully negative dependency can't actually be if there's There's three things, you know if one is the exact opposite of the other then what is the third thing? So it only exists in the bivariate case So we can kind of see you know if f of x is equal to 4 then f of y has to be equal to 2 And this is going to be equal to 1 of the 6 chance because it means you're rolling 4 on the dice I remember y is simply going to be x or 6 minus x in this example So this is kind of the boundary copula copulas We can make them a little bit complicated when we start combining them all together So we can combine the Independent the minimum and the maximum all together in this very big formula over here As you can see we've got p we've got q and then 1 minus p minus q is the independent and Of course these values just have to meet certain Requirements that both have to be between 0 and 1 and together they can't be greater than 1 But what is fascinating is how these things now relate to our correlation? I mean we've got Spearman's row is equal to q minus p and Kendall's tau is going to be equal to q minus p times 2 plus p plus q Divided by 3 so we can't take all the boundary copulas Together and we can combine them into this formula over here Which we can then use to try and represent other copulas of course this mathematically does get a little bit Complicated so in the next video we are going to be look at the Archimedean copulas, which are a little bit more Elegant so I'll see you guys there, but like I said, thanks so much for watching. This has been the boundary copulas