How are marginal distributions found from a joint probability distribution?

In probability theory, a marginal distribution is derived from a joint distribution by summing the probabilities of the joint outcomes over all possible values of the other variable.

If you have two random variables, X and Y, with a joint distribution function $h(x_i,y_j)=P\{X=x_i,Y=y_j\}$, the marginal distribution of X, denoted as $f(x_i)$, is found by summing the joint probabilities across all values of Y:

$$f(x_i)=\sum _{j}h(x_i,y_j)$$

Similarly, the marginal distribution of Y, denoted as $g(y_j)$, is found by summing the joint probabilities across all values of X:

$$g(y_j)=\sum _{i}h(x_i,y_j)$$

This process effectively "marginalizes" out one variable to focus on the individual probability distribution of the other.1