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Consider a random vector $(X,Y,Z)$, Let $f_X, f_Y, f_Z$ be the probability distributions of each component.

Question: Does there always exist a distribution $f$ for the whole vector $(X,Y,Z)$ such that:

  • The marginals of $f$ are $f_X, f_Y, f_Z$

  • $X,Y,Z$ are mutually independent random variables under $f$

Thoughts: In my opinion, the answer is yes. This is by applying Sklar's Theorem with the independence copula. If this is the case, could you formalise it better?

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  • $\begingroup$ My thought is the same as yours. $\endgroup$ Jan 25, 2022 at 8:08
  • $\begingroup$ Isn't this just the existence of the product measure or am I missing something? $\endgroup$
    – g g
    Jan 26, 2022 at 12:18

1 Answer 1

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You have already specified that the given probability distributions are the marginal distributions of the elements of the vector, so the first condition holds by assumption. Mutual independence is obtained by taking the product measure for the joint distribution, which gives the joint density:

$$f_{X,Y,Z}(x,y,z) = f_X(x) \cdot f_Y(y) \cdot f_Z(z).$$

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