Legitimate probability mass function

In this lecture we analyze two properties of probability mass functions (pmfs).

We prove not only that any probability mass function satisfies these two properties, but also that any function satisfying them is a legitimate pmf.

As a consequence, when we need to check whether a function is a valid pmf, we just need to verify that the two properties hold.

Table of contents

Properties of probability mass functions
How to verify that a pmf is valid
How to build valid pmfs
Solved exercises

Properties of probability mass functions

Let us start with a formal characterization.

Proposition Let be a discrete random variable and let be its probability mass function. Then, the function satisfies the following two properties:

Non-negativity: for any ;
Sum over the support equals : , where is the support of .

Proof

Remember that, by the definition of a probability mass function, is such that [eq6]

Probabilities cannot be negative, therefore and, as a consequence, . This proves Property 1 (non-negativity).

Furthermore, the probability of a sure thing must be equal to . Since, by the very definition of support, the event is a sure thing, then [eq10] which proves Property 2 (sum over the support equals ).

How to verify that a pmf is valid

Any probability mass function must satisfy Properties 1 and 2 above.

By using some standard results from measure theory (omitted here), it is possible to prove that the converse is also true: any function satisfying the two properties above is a pmf.

Proposition Let be a function satisfying the following two properties:

Non-negativity: for any ;
Sum over the support equals : , where is the support of .

Then, there exists a discrete random variable whose probability mass function is .

As a consequence, we only need to check that these two properties hold when we want to prove that a function is a valid pmf.

How to build valid pmfs

The proposition above gives us a powerful method for constructing probability mass functions.

Take a subset of the set of real numbers .

Take any function that is non-negative on (non-negative means that for any $xin R_{X}$ ).

If the sum [eq18] is well-defined and is finite and strictly positive, then define

Since is strictly positive, is non-negative and it satisfies Property 1.

The function also satisfies Property 2 because [eq22]

Therefore, any function that is non-negative on ( is chosen arbitrarily) can be used to construct a pmf if its sum over is well-defined and it is finite and strictly positive.

Example Defineand a function as follows: [eq24] Can we use to build a probability mass function? First of all, we have to check that is non-negative. This is obviously true, because $x^{2}$ is always non-negative. Then, we have to check that the sum of over exists and is finite and strictly positive: [eq25] Since exists and is finite and strictly positive, we can define [eq26] By the above proposition, is a valid probability mass function.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Consider the following function: [eq28]

Determine whether is a valid probability mass function.

Solution

For we havewhile for we haveTherefore, for any and the non-negativity property is satisfied. The other necessary property (sum over the support equals ) is also satisfied because [eq35]

Exercise 2

Consider the following function: [eq36]

Check that is a legitimate probability mass function.

Solution

For we havewhile for we haveTherefore, for any and the non-negativity property is satisfied. The other necessary property (sum over the support equals ) is also satisfied because [eq43]

Exercise 3

Consider the following function: [eq44]

Prove that is a valid pmf.

Solution

For we havebecause $4^{1-x}$ is strictly positive. For we haveTherefore, for any and the non-negativity property is satisfied. The other necessary property (sum over the support equals ) is also satisfied because [eq49]

How to cite

Please cite as:

Taboga, Marco (2021). "Legitimate probability mass function", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/legitimate-probability-mass-function.