 Hi everyone, it's MJ, the fellow actor, and in this video I want to talk about Sklar's Theorem. Now, I hope I'm pronouncing his name correctly, Abe Sklar. So he was an American who basically put forward this theorem in 1959. What he did is he wrote a letter to a very prominent statistician, Maurice René Frichet. I don't know if I'm pronouncing the French properly, who was doing a lot of work around joint distributions in 1956. Sklar writes to him in 1959, Maurice polishes up a little bit of his French, changes one or two things, and then publishes this as Sklar's Theorem in a French Journal. And essentially the theorem is as follows. It says, let f of x and f of y be marginal cumulative distribution functions of two random variables x and y, which means f of x is equal to the probability that x is less than an x and the f of y is equal to the probability that y is less of y. Then it says, let fx of y be a joint cumulative distribution function such that fx of y is equal to the probability of x less than x, given that y is less than y. And then here comes the magic. It says, then fx of y is linked to f of x and f of y through a copula or the joint distribution is linked to the marginal distributions through a copula. And thus we get this basic formula or this framework for the copulas, which says the fx of y is equal to this copula function, which is taking two marginal distributions as its input. Now, if you're writing the actual exam and they ask you to state Sklar's Theorem, do not forget the following point, because this is going to get you your final half mark. And that is to say that because Sklar also said, and if f of x and f of y are continuous, then the copula is going to be unique. Anyway, that is Sklar's Theorem. Thank you so much for watching.