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.
The following is a formal definition.
Definition Let and be two discrete random variables. The conditional probability mass function of given is a function such thatfor any , where is the conditional probability that , given that .
In order to derive the conditional pmf of a discrete variable given the realization of another discrete variable , we need to know their joint probability mass function .
Suppose that we are informed that , where denotes the value taken by (called the realization of ).
How do we take this information into account? By deriving the conditional probability mass function of .
The derivation involves two steps:
first, we compute the marginal probability mass function of by summing the joint probability mass over the support of (i.e., the set of all its possible values, denoted by ):
then, we compute the conditional pmf as follows:
Here is an example.
Take two discrete variables and and consider them jointly as a random vector
Suppose that the support of this vector is and that its joint pmf is
Let us compute the conditional pmf of given .
The support of is
The marginal pmf of evaluated at is
The support of is
Therefore, the conditional pmf of conditional on is
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 and the conditional , then we can multiply them and obtain the joint distribution:
You can find more details about the conditional probability mass function in the lecture entitled Conditional probability distributions.
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