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Probability mass function

by , PhD

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.

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In formal terms, the probability mass function of a discrete random variable X is a function [eq1] such that[eq2]where [eq3] is the probability that the realization of the random variable X will be equal to x.


Suppose a random variable X can take only three values (1, 2 and 3), each with equal probability. Its probability mass function is[eq4]

So, for example,[eq5]that is, the probability that X will be equal to $2$ is $1/3$. Or, [eq6]that is, the probability that X will be equal to $3/5$ is equal to 0.


Note that the probability mass function is defined on all of R, 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 $X $ (i.e., to the set of values that the variable X can take). On the contrary, the value of the pmf is positive for the arguments that belong to the support of X.

In the example above, the support of X is [eq7]As a consequence, the pmf is positive on the support R_X 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]We set the parameter $lambda =2$ and plot the values of the pmf only for arguments smaller than $10$ (note that the support of the distribution is $U{2124} _{+}$, i.e., the set of all non-negative integers, but the values of [eq9] become very small for $x>10$).

More details

You can find an in-depth discussion of probability mass functions in the lecture entitled Random variables.

Related concepts

Related concepts can be found in the following glossary entries:

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How to cite

Please cite as:

Taboga, Marco (2021). "Probability mass function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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