6

Random variable X and Y with cumulative distribution functions $ F_X(x)$ and $F_Y(y)$,are independent iff the combined random variable $(X, Y)$ has a joint cumulative distribution function $$F_{X,Y}(x,y) = F_X(x) F_Y(y)$$

In the above definition,they use joint distribution of $(X,Y)$. According to the definition of joint probability distribution of random vector such as $(X,Y)$, $X$ and $Y$ must be defined on the same probability space. Does it mean $X$ and $Y$ when defining independence should be on the same probability space?

relate question:Can we define a joint probability distribution over different sample spaces / probability spaces?

Spaceship222
  • 479
  • 6
    Yes, they must be on the same probability space. It is the only way for $P[X\leq x, Y\leq y]$ to make sense. It is $$P[X\leq x, Y\leq y] = P[{\omega \in \Omega : X(\omega)\leq x \mbox{ and } Y(\omega)\leq y}]$$ – Michael Jun 14 '18 at 03:47
  • You can also see this fact defining each of your variables on the product measure space. In this case, there will be no way to construct dependent variables – Ariel Serranoni Jun 14 '18 at 03:52
  • @Michael thanks.Does it implicitly assume they are on the same probability space when considering relation of random variables or events? – Spaceship222 Jun 14 '18 at 04:09
  • 1
    @Spaceship222 Yes. If there are two events or random variables that you're talking about in the same problem/context, they are most likely on the same space. Certainly in order to make sense of a joint relationship (even if that relationship is independence), they need to be on the same space. – spaceisdarkgreen Jun 14 '18 at 04:26

0 Answers0