It's easy to think about multiple finite discrete random variables that are independent or correlated. Say two binary variables that are in uniform distribution by themselves. In the extreme cases they may always take the same or opposite values. The compete possibilities could be described by a 2 by 2 matrix, but they could be visualized like something between the same and opposite cases.
But how could I imagine two exponential distributions being correlated? (Continuous or discrete doesn't matter. Just answer with whichever is simplest.) While it still works by listing an infinite matrix (or a 2d function, or anything similar), it seems too arbitrary, and doesn't seems to help me understanding anything. Are there, say, some named parameters that could describe some general patterns of some distinctive correlations?
If I have to ask one very concrete question, the question is, exponential distributions could have the same correlations in some specific value pairs like the random variables taking values in a finite range, but are they exactly that? Are there anything more I need to notice because of the values are unbounded?
(For the purpose of this question, I'm thinking about a human designing something with perfect exponential distribution, so every extreme and impractical cases, such as zero probability not equivalent to impossible, are considered, not just something tested to have exponential distribution. This question is mainly about the exponential distribution, but if there are theories that could be applied to any distributions they could also be helpful.)