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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 X be a discrete random variable and let [eq1] be its probability mass function. The probability mass function [eq2] satisfies the following two properties:

  1. Non-negativity: [eq3] for any $xin U{211d} $;

  2. Sum over the support equals 1: [eq4], where R_X is the support of X.

Proof

Remember that, by the definition of a probability mass function, [eq5] is such that[eq6]

Probabilities cannot be negative, therefore [eq7] and, as a consequence, [eq3]. This proves property 1 above (non-negativity).

Furthermore, the probability of a sure thing must be equal to 1. Since, by the very definition of support, the event [eq9] is a sure thing, then[eq10]which proves property 2 above (sum over the support equals 1).

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 [eq11] satisfying the two properties above is a probability mass function.

Proposition (Legitimate probability mass function) Let [eq2] be a function satisfying the following two properties:

  1. Non-negativity: [eq3] for any $xin U{211d} $;

  2. Sum over the support equals 1: [eq4], where R_X is the support of X.

Then, there exists a discrete random variable X whose probability mass function is [eq2].

This proposition gives us a powerful method for constructing probability mass functions. Take a subset of the set of real numbers [eq16]. Take any function g(x) that is non-negative on R_X (non-negative means that [eq17] for any $xin R_{X}$). If the sum[eq18]is well-defined and is finite and strictly positive, then define[eq19]$S$ is strictly positive, thus [eq2] is non-negative and it satisfies property 1. It also satisfies Property 2 because[eq21]Therefore, any function g(x) that is non-negative on R_X (R_X is chosen arbitrarily) can be used to construct a probability mass function if its sum over R_X is well-defined and is finite and strictly positive.

Example Define[eq22]and a function g(x) as follows:[eq23]Can we use g(x) to build a probability mass function? First of all, we have to check that g(x) is non-negative. This is obviously true, because $x^{2}$ is always non-negative. Then, we have to check that the sum of g(x) over R_X exists and is finite and strictly positive:[eq24]Since $S$ exists and is finite and strictly positive, we can define[eq25]By the above proposition, [eq2] is a legitimate probability mass function.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Consider the following function:[eq27]

Prove that [eq2] is a legitimate probability mass function.

Solution

For [eq29] we have[eq30]while for [eq31] we have[eq32]Therefore, [eq3] for any $xin U{211d} $ and the non-negativity property is satisfied. The other necessary property (sum over the support equals 1) is also satisfied because[eq34]

Exercise 2

Consider the following function:[eq35]

Prove that [eq2] is a legitimate probability mass function.

Solution

For [eq37] we have[eq38]while for [eq39] we have[eq32]Therefore, [eq3] for any $xin U{211d} $ and the non-negativity property is satisfied. The other necessary property (sum over the support equals 1) is also satisfied because[eq42]

Exercise 3

Consider the following function:[eq43]

Prove that [eq2] is a legitimate probability mass function.

Solution

For $xin U{2115} $ we have[eq45]because $4^{1-x}$ is strictly positive. For $x
otin U{2115} $ we have[eq32]Therefore, [eq3] for any $xin U{211d} $ and the non-negativity property is satisfied. The other necessary property (sum over the support equals 1) is also satisfied because[eq48]

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