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.
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 .
Remember that, by the definition of a probability mass function, is such that
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, thenwhich proves Property 2 (sum over the support equals ).
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.
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 ).
If the sumis 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
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: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 is always non-negative. Then, we have to check that the sum of over exists and is finite and strictly positive:Since exists and is finite and strictly positive, we can defineBy the above proposition, is a valid probability mass function.
Below you can find some exercises with explained solutions.
Consider the following function:
Determine whether is a valid probability mass function.
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
Consider the following function:
Check that is a legitimate probability mass function.
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
Consider the following function:
Prove that is a valid pmf.
For we havebecause 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
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.
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