# Marginal distribution

In probability theory, given a joint probability density function of two parameters *x* and *y*, the **marginal distribution** of *x* is the probability distribution of *x* after information about *y* has been averaged over. From a Bayesian probability perspective, we can consider the joint probability density as a joint inference about the true values of the two parameters and the marginal distribution of (say) *x*, as our inference about *x* after the uncertainty about *y* had been averaged over. We can say that in this case, we are considering *y* as a nuisance parameter.

For continuous probability densities, this marginal probability density function can be written as *m*_{y}(*x*). Such that

where *p*(*x*,*y*) gives the joint distribution of *x* and *y*, and *c*(*x*|*y*) gives the conditional distribution for *x* given *y*. Note that the marginal distribution has the form of an expectation.

For a discrete probability mass function, the marginal probability for the k^{th} ordinate can be written as *p*_{k} Such that

where the *j* index spans all values of the discrete *y*. With *k* fixed here and *p*_{k,j} considered as a matrix, then this can be thought of as summing over all columns in the k^{th} row. Similarly, the marginal mass function for *y* can be computed by summing over all rows in a particular column. When all of the *p*_{k} are determined this way for all k, this set of *p*_{k} constitute the discrete probability mass function for the relevant discrete values of *x*, in this particular case calculated as a marginal mass function from an original joint probability mass function.