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Marginal probability mass function

Consider a discrete random vector, that is, a vector whose entries are discrete random variables. When one of these entries is taken in isolation, its distribution can be characterized in terms of its probability mass function. This is called marginal probability mass function, in order to distinguish it from the joint probability mass function, which is instead used to characterize the joint distribution of all the entries of the random vector considered together.

Definition

The following is a more formal definition.

Definition Let [eq1] be K discrete random variables forming a Kx1 random vector. Then, for each $i=1,ldots ,K$, the probability mass function of the random variable X_i, denoted by [eq2], is called marginal probability mass function.

Remember that the probability mass function [eq3] is a function [eq4] such that[eq5]where [eq6] is the probability that X_i will be equal to x.

By contrast, the joint probability mass function of the vector X is a function [eq7] such that[eq8]where [eq9] is the probability that X_i will be equal to $x_{i}$, simultaneously for all $i=1,ldots ,K$.

How to derive it

Denote by R_X the support of X (i.e., the set of all values it can take). The marginal probability mass function of X_i is obtained from the joint probability mass function as follows:[eq10]where the sum is over the set[eq11]In other words, the marginal probability mass function of X_i at the point x is obtained by summing the joint probability mass function $p_{X}$ over all the vectors that belong to the support R_X and are such that their i-th component is equal to x.

Example

Let X be a $2	imes 1$ random vector with support[eq12]and joint probability mass function[eq13]

The marginal probability mass function of X_1 evaluated at the point $x_{1}=1$ is[eq14]

When evaluated at the point $x_{1}=2$ it is[eq15]

For all the other points, it is equal to zero. Therefore, we have[eq16]

More details

A more detailed discussion of the marginal probability mass function can be found in the lecture entitled Random vectors.

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