How does a joint distribution describe two random variables at the same time?

In the context of probability and statistics, a joint distribution describes the relationship between two or more random variables defined on the same sample space by assigning probabilities to the simultaneous occurrence of their specific values.1

For two discrete random variables, $X$ and $Y$, the joint distribution is represented by a function $h(x_i,y_j)$, which defines the probability that $X$ takes the value $x_i$ and $Y$ takes the value $y_j$ at the same time:
$$h(x_i,y_j)=P\{X=x_i,Y=y_j\}$$

This function allows for the analysis of how the variables interact. From the joint distribution, one can derive the individual behavior of each variable through marginal distributions. For instance, the marginal distribution of $X$ is found by summing the joint probabilities over all possible values of $Y$:
$$f(x_i)=\sum _{j}h(x_i,y_j)$$

If the variables are independent, the joint distribution is simply the product of their individual marginal distributions, meaning $h(x_i,y_j)=f(x_i)g(y_j)$ for all values.1