Probability Mass Functions

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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.

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

(1) P(X = x) = f(x) > 0  if x ∈ the support S

(2)  \(\sum\limits_{x\in S} f(x)=1\)

(3) \(P(X\in A)=\sum\limits_{x\in A} f(x)\)

Item #1 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. Item #2 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, item #3 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

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

Let f(x) = cx2 for x = 1, 2, 3. Determine the constant c so that the function f(x) satisfies the conditions of being a probability mass function.

Solution. 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

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, ...

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