A marginal distribution is a probability distribution of a subset of a collection of random variables, showing the probabilities of various values of the variables in the subset without reference to the values of the other variables. It contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out.
In simpler terms, a marginal distribution is where you are only interested in one of the random variables, ignoring the other related variables in a dataset. It can be found by summing the joint probability distribution over all values of the variables not of interest. For example, given a known joint distribution of two discrete random variables, the marginal distribution of one variable is the probability distribution of that variable when the values of the other variable are not taken into consideration.
Marginal distributions are often found using contingency tables, where the values in the margins represent the distribution of one variable without considering the other variable. These distributions play an important role in the characterization of independence between random variables, as two random variables are independent if and only if their joint distribution function is equal to the product of their marginal distribution functions.
In summary, a marginal distribution provides the probability distribution of a single variable in a subset of variables, without considering the values of the other variables, and it is an essential concept in probability theory and statistics.
