The probability mass function (pmf) characterizes the distribution of a discrete random variable. It associates to any given number the probability that the random variable will be equal to that number.
In formal terms, the probability mass function of a discrete random variable
is a function
![[eq1]](/images/probability-mass-function__2.png){kind=small}
such
that![[eq2]](/images/probability-mass-function__3.png){kind=small}
where
![[eq3]](/images/probability-mass-function__4.png){kind=small}
is the probability that the realization of the random variable
will be equal to
.
Suppose a random variable
can take only three values (1, 2 and 3), each with equal probability. Its
probability mass function
is![[eq4]](/images/probability-mass-function__8.png)
So, for
example,![[eq5]](/images/probability-mass-function__9.png){kind=small}
that
is, the probability that
will be equal to
is
.
Or,
![[eq6]](/images/probability-mass-function__13.png){kind=small}
that
is, the probability that
will be equal to
is equal to
.
Note that the probability mass function is defined on all of
,
that is, it can take as argument any real number. However, its value is equal
to zero for all those arguments that do not belong to the support of
(i.e., to the set of values that the variable
can take). On the contrary, the value of the pmf is positive for the arguments
that belong to the support of
.
In the example above, the support of
is
![[eq7]](/images/probability-mass-function__22.png){kind=small}
As
a consequence, the pmf is positive on the support
and equal to zero everywhere else.
Often, probability mass functions are plotted as column charts. For example,
the following plot shows the pmf of the
Poisson
distribution, which
is![[eq8]](/images/probability-mass-function__24.png)
We
set the parameter
and plot the values of the pmf only for arguments smaller than
(note that the support of the distribution is
,
i.e., the set of all non-negative integers, but the values of
become very small for
).
You can find an in-depth discussion of probability mass functions in the lecture entitled Random variables.
Related concepts can be found in the following glossary entries:
Joint probability mass function: the pmf of a random vector.
Marginal probability mass function: the pmf obtained by considering only a subset of the set of random variables forming a given random vector.
Conditional probability mass function: the pmf obtained by conditioning on the realization of another random variable.
Previous entry: Probability density function
Next entry: Probability space
Please cite as:
Taboga, Marco (2021). "Probability mass function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/probability-mass-function.
Most of the learning materials found on this website are now available in a traditional textbook format.
