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Conditional probability mass function

by , PhD

The probability distribution of a discrete random variable can be characterized by its probability mass function (pmf). When the probability distribution of the random variable is updated, in order to consider some information that gives rise to a conditional probability distribution, then such a conditional distribution can be characterized by a conditional probability mass function.

Table of Contents

Definition

The following is a formal definition.

Definition Let X and Y be two discrete random variables. The conditional probability mass function of X given Y=y is a function [eq1] such that[eq2]for any $xin U{211d} $, where [eq3] is the conditional probability that $X=x$, given that Y=y.

How to compute the conditional pmf

In order to derive the conditional pmf of a discrete variable X given the realization of another discrete variable Y, we need to know their joint probability mass function [eq4].

Suppose that we are informed that Y=y, where $y$ denotes the value taken by Y (called the realization of Y).

How do we take this information into account? By deriving the conditional probability mass function of X.

The derivation involves two steps:

  1. first, we compute the marginal probability mass function of Y by summing the joint probability mass over the support of X (i.e., the set of all its possible values, denoted by R_X):[eq5]

  2. then, we compute the conditional pmf as follows:[eq6]

An example

Here is an example.

Take two discrete variables X and Y and consider them jointly as a random vector[eq7]

Suppose that the support of this vector is [eq8]and that its joint pmf is[eq9]

Let us compute the conditional pmf of X given $Y=0$.

The support of Y is[eq10]

The marginal pmf of Y evaluated at $y=0$ is[eq11]

The support of X is[eq12]

Therefore, the conditional pmf of X conditional on $Y=0$ is[eq13]

How to derive the joint pmf from the conditional and marginal

The previous example showed how the conditional pmf can be derived from the joint pmf. We can easily do the other way around.

If we know the marginal pmf [eq14] and the conditional [eq15], then we can multiply them and obtain the joint distribution:[eq16]

More details

You can find more details about the conditional probability mass function in the lecture entitled Conditional probability distributions.

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