# #StackBounty: #probability #distributions #random-variable From trivariate cdf to the distribution of differences of random variables

### Bounty: 50

Consider a trivariate cumulative distribution function (cdf) $$G$$.

• Is there a collection of necessary conditions on $$G$$ ensuring that
$$exists text{ a random vector (X_1,X_2) such that (X_1, X_2, X_1-X_2) has cdf G}$$
?

• Is there a collection of necessary and sufficient conditions on $$G$$ ensuring that
$$exists text{ a random vector (X_1,X_2) such that (X_1, X_2, X_1-X_2) has cdf G}$$
?

Update: Let $$P$$ be the probability distribution associated with $$G$$. We can claim that: if there exists a random vector $$(X_1,X_2)$$ such that $$(X_1, X_2, X_1-X_2)$$ has probability distribution $$P$$, then
$$int_{(a,b,c)in mathbb{R}^3 text{ s.t. } c=a-b} dP=1$$

• Is this condition also sufficient? I.e., can we claim that if
$$int_{(a,b,c)in mathbb{R}^3 text{ s.t. } c=a-b} dP=1$$
then
there exists a random vector $$(X_1,X_2)$$ such that $$(X_1, X_2, X_1-X_2)$$ has probability distribution $$P$$?

• Can we write
$$int_{(a,b,c)in mathbb{R}^3 text{ s.t. } c=a-b} dP=1$$
by using the cdf $$G$$
?

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