 StatLect

Legitimate probability mass functions

In this lecture we analyze two properties of probability mass functions. We prove not only that any probability mass function satisfies these two properties, but also that any function satisfying these two properties is a legitimate probability mass function. Properties of probability mass functions

Any probability mass function satisfies two basic properties, as shown by the next proposition.

Proposition (Properties of a probability mass function) Let be a discrete random variable and let be its probability mass function. The probability mass function satisfies the following two properties:

1. Non-negativity: for any ;

2. Sum over the support equals : , where is the support of .

Proof

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 above (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 which proves property 2 above (sum over the support equals ).

Identification of legitimate probability mass function

Any probability mass function must satisfy property 1 and 2 above. Using some standard results from measure theory (omitted here), it is possible to prove that the converse is also true, that is, any function satisfying the two properties above is a probability mass function.

Proposition (Legitimate probability mass function) Let be a function satisfying the following two properties:

1. Non-negativity: for any ;

2. Sum over the support equals : , where is the support of .

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

This proposition 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 sum is well-defined and is finite and strictly positive, then define  is strictly positive, thus is non-negative and it satisfies property 1. It also satisfies Property 2 because Therefore, any function that is non-negative on ( is chosen arbitrarily) can be used to construct a probability mass function if its sum over is well-defined and is finite and strictly positive.

Example Define and 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 define By the above proposition, is a legitimate probability mass function.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Consider the following function: Prove that is a legitimate probability mass function.

Solution

For we have while for we have Therefore, for any and the non-negativity property is satisfied. The other necessary property (sum over the support equals ) is also satisfied because Exercise 2

Consider the following function: Prove that is a legitimate probability mass function.

Solution

For we have while for we have Therefore, for any and the non-negativity property is satisfied. The other necessary property (sum over the support equals ) is also satisfied because Exercise 3

Consider the following function: Prove that is a legitimate probability mass function.

Solution

For we have because is strictly positive. For we have Therefore, for any and the non-negativity property is satisfied. The other necessary property (sum over the support equals ) is also satisfied because The book

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