Properties of the probability mass function¶

There are some standard properties of the probability mass function that are worth memorizing.

First, he probability mass function is nonnegative:

\[ p(x) \ge 0, \]

for the possible values \(x\) that the random variable \(X\) can take.

Second, the probability mass function is normalized:

\[ \sum_x p(x) = 1. \]

The sum is over all the possible values of \(X\). This is a direct consequence of the fact that \(X\) must take some value of all that are possible.

Finally, you can use the probability mass function to find the probability that \(X\) takes values in any set of possibilities. For example:

\[ p(X=x_1\;\text{or}\;X=x_2) = p(x_1) + p(x_2), \]

if \(x_1\not= x_2\).

Similarly, if \(x_1, x_2,\) and \(x_3\) are all different, then:

\[ p(X=x_1\;\text{or}\;X=x_2\;\text{or}\;X=x_3) = p(x_1) + p(x_2) + p(x_3). \]

An, in general for any set of possible values of \(X\), say \(A\), the probability of \(X\) taking values in \(A\) is:

\[ p(X\in A) = \sum_{x\in A} p(x). \]

Introduction to Data Science for Mechanical Engineers (Lecture Book)