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for all joint distributions satisfying.
for all joint distributions of the form (17). Proof.
We combine these three components into joint distributions to obtain completed cloud representation.
In this study, the joint distributions of order statistics of innid discrete random variables are expressed.
Also, the joint distributions are obtained in the form of an integral.
where,,, and are auxiliary random variables, and the maximum is over all joint distributions satisfying.
Hence, for the outer bound, it is sufficient to consider the joint distributions having the form.
Both marginal and joint distributions are derived using the entropy theory.
Copula function is a bridge connecting the marginal and joint distributions of the random variables [21].
Fig. 3. Joint distributions of the predictive parameters at depths of 10 and 25 km.
This avoids directly comparing the joint distributions of computational and experimental responses.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com