How is a joint distribution used to describe two random variables from the same experiment?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, when two random variables, $X$ and $Y$ , are defined on the same sample space, their relationship is described using a joint distribution.1

The joint distribution is defined as a function of two variables, $h (x_{i}, y_{j}) = P {X = x_{i}, Y = y_{j}}$ , which represents the probability that $X$ assumes the value $x_{i}$ and $Y$ assumes the value $y_{j}$ simultaneously.1

This framework allows for the derivation of individual behaviors through marginal distributions:
* The marginal distribution of $X$ , denoted as $f (x_{i})$ , is found by summing the joint probabilities over all possible values of $Y$ .1
* The marginal distribution of $Y$ , denoted as $g (y_{j})$ , is found by summing the joint probabilities over all possible values of $X$ .1

Furthermore, the joint distribution is used to determine if $X$ and $Y$ are independent; they are considered independent if $h (x_{i}, y_{j}) = f (x_{i}) g (y_{j})$ for all possible values of $i$ and $j$ .1