7.2 - Probability Mass Functions

7.2 - Probability Mass Functions

The probability that a discrete random variable $$X$$ takes on a particular value $$x$$, that is, $$P(X = x)$$, is frequently denoted $$f(x)$$. The function $$f(x)$$ is typically called the probability mass function, although some authors also refer to it as the probability function, the frequency function, or probability density function. We will use the common terminology — the probability mass function — and its common abbreviation —the p.m.f.

Probability Mass Function

The probability mass function, $$P(X=x)=f(x)$$, of a discrete random variable $$X$$ is a function that satisfies the following properties:

• $$P(X=x)=f(x)>0$$, if $$x\in \text{ the support }S$$
• $$\sum\limits_{x\in S} f(x)=1$$
• $$P(X\in A)=\sum\limits_{x\in A} f(x)$$

First item basically says that, for every element $$x$$ in the support $$S$$, all of the probabilities must be positive. Note that if $$x$$ does not belong in the support $$S$$, then $$f(x) = 0$$. The second item basically says that if you add up the probabilities for all of the possible $$x$$ values in the support $$S$$, then the sum must equal 1. And, the third item says to determine the probability associated with the event $$A$$, you just sum up the probabilities of the $$x$$ values in $$A$$.

Since $$f(x)$$ is a function, it can be presented:

• in tabular form
• in graphical form
• as a formula

Let's take a look at a few examples.

Example 7-4

Let $$X$$ equal the number of siblings of Penn State students. The support of $$X$$ is, of course, 0, 1, 2, 3, ... Because the support contains a countably infinite number of possible values, $$X$$ is a discrete random variable with a probability mass function. Find $$f(x) = P(X = x)$$, the probability mass function of $$X$$, for all $$x$$ in the support.

This example illustrated the tabular and graphical forms of a p.m.f. Now let's take a look at an example of a p.m.f. in functional form.

Example 7-5

Let $$f(x)=cx^2$$ for $$x = 1, 2, 3$$. Determine the constant $$c$$ so that the function $$f(x)$$ satisfies the conditions of being a probability mass function.

The key to finding $$c$$ is to use item #2 in the definition of a p.m.f.

The support in this example is finite. Let's take a look at an example in which the support is countably infinite.

Example 7-6

Determine the constant $$c$$ so that the following p.m.f. of the random variable $$Y$$ is a valid probability mass function:

$$f(y)=c\left(\dfrac{1}{4}\right)^y$$ for y = 1, 2, 3, ...

Again, the key to finding $$c$$ is to use item #2 in the definition of a p.m.f.