How are marginal distributions found from a joint distribution?

To find the marginal distribution of a random variable from a joint distribution, you sum the joint probabilities over all possible values of the other variable.

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

$$f(x_i)=\sum _{j}h(x_i,y_j)$$1

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

$$g(y_j)=\sum _{i}h(x_i,y_j)$$1

This process effectively "marginalizes" out one variable to isolate the probability distribution of the other. The term "marginal" is used to emphasize the relationship between the individual distribution and the joint distribution of the variables involved.1