How can the sum of two dice and the number of even dice be studied with a joint distribution?

To study the sum of two dice and the number of even dice using a joint distribution, you define two random variables on the same sample space.1

Let $X$ represent the sum of the numbers appearing on the two dice, where $X$ can take values from 2 to 12. Let $Y$ represent the number of even numbers appearing, which can take values of 0, 1, or 2.1

The joint distribution is defined by the function $h(x_i,y_j)=P\{X=x_i,Y=y_j\}$, which calculates the probability of both events occurring simultaneously for all possible pairs of $i$ and $j$. Because there are 36 equally likely outcomes when throwing two dice, this distribution is determined through direct enumeration of the outcomes that satisfy both conditions.1

From this joint distribution, you can also derive:
* Marginal distributions: The individual probability distributions for $X$ or $Y$ alone, calculated by summing the joint probabilities over the other variable.1
* Independence: If $h(x_i,y_j)=f(x_i)g(y_j)$ for all $i$ and $j$, the variables are considered independent.1