The marginal probability distribution of a random variable that has a joint probability distribution with some other random variables is the probability distribution of that random variable without regard for the values that the other random variables take. The marginal distribution of a discrete random variable X₁ that has a joint distribution with other discrete random variables can be found from the joint distribution by summing over all possible values of the other variables. For example, suppose we roll two fair dice independently. Let X₁ be the number of spots that show on the first die, and let X₂ be the total number of spots that show on both dice. Then the joint distribution of X₁ and X₂ is as follows:

P(X₁ = 1, X₂ = 2) = P(X₁ = 1, X₂ = 3) = P(X₁ = 1, X₂ = 4) = P(X₁ = 1, X₂ = 5) = P(X₁ = 1, X₂ = 6) = P(X₁ = 1, X₂ = 7) =

P(X₁ = 2, X₂ = 3) = P(X₁ = 2, X₂ = 4) = P(X₁ = 2, X₂ = 5) = P(X₁ = 2, X₂ = 6) = P(X₁ = 2, X₂ = 7) = P(X₁ = 2, X₂ = 8) = …

… P(X₁ = 6, X₂ = 7) = P(X₁ = 6, X₂ = 8) = P(X₁ = 6, X₂ = 9) = P(X₁ = 6, X₂ = 10) = P(X₁ = 6, X₂ = 11) = P(X₁ = 6, X₂ = 12) = 1/36.

The marginal probability distribution of X₁ is

P(X₁ = 1) = P(X₁ = 2) = P(X₁ = 3) = P(X₁ = 4) = P(X₁ = 5) = P(X₁ = 6) = 1/6.

We can verify that the marginal probability that X₁ = 1 is indeed the sum of the joint probability distribution over all possible values of X₂ for which X₁ = 1:

P(X₁ = 1) = P(X₁ = 1, X₂ = 2) + P(X₁ = 1, X₂ = 3) + P(X₁ = 1, X₂ = 4) + P(X₁ = 1, X₂ = 5) + P(X₁ = 1, X₂ = 6) + P(X₁ = 1, X₂ = 7) = 6/36 = 1/6.

Similarly, the marginal probability distribution of X₂ is

P(X₂ = 2) = P(X₂ = 12) = 1/36

P(X₂ = 3) = P(X₂ = 11) = 1/18

P(X₂ = 4) = P(X₂ = 10) = 3/36

P(X₂ = 5) = P(X₂ = 9) = 1/9

P(X₂ = 6) = P(X₂ = 8) = 5/36

P(X₂ = 7) = 1/6.

Again, we can verify that the marginal probability that X₂ = 4 is 3/36 by adding the joint probabilities for all possible values of X₁ for which X₂ = 4:

P(X₂ = 4) = P(X₁ = 1, X₂ = 4) + P(X₁ = 2, X₂ = 4) + P(X₁ = 3, X₂ = 4) = 3/36.