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.
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:
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 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, thenwhich proves property 2 above (sum over the support equals ).
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:
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 .
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 sumis 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 becauseTherefore, 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 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 legitimate probability mass function.
Below you can find some exercises with explained solutions.
Consider the following function:
Prove 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 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 legitimate probability mass function.
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
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