Introduction

In the realm of probability and statistics, understanding the nuances between a Probability Mass Function (PMF) and a Probability Density Function (PDF) is crucial. These functions serve as fundamental tools to describe the behavior of random variables, whether discrete or continuous. This chapter delves into the distinctions between PMF and PDF, shedding light on their applications, interpretations, and the types of variables they represent.

Probability distributions are essential for comprehending the likelihood of different outcomes in random experiments. The choice between a PMF or a PDF depends on whether the random variable is discrete or continuous. Both functions convey valuable information about the probabilities associated with specific values or intervals of a random variable.

Probability Mass Function (PMF)

A Probability Mass Function (PMF) is primarily associated with discrete random variables. It assigns probabilities to each possible value that the random variable can take. In other words, for a discrete random variable, the PMF provides a detailed mapping of the probabilities of occurrence for each distinct outcome.

Mathematically, the PMF of a discrete random variable $X$ is denoted as $P(X=x)$, where ‘$x$’ represents a specific value that $X$ can take. The PMF satisfies two key properties: each probability is non-negative, and the sum of all probabilities is equal to 1.

Probability Density Function (PDF)

In contrast, a Probability Density Function (PDF) is used for continuous random variables. Instead of assigning probabilities to specific values, the PDF describes the likelihood of a random variable falling within a certain interval. The area under the PDF curve over a given interval corresponds to the probability that the random variable falls within that interval.

For a continuous random variable $X$, the $PDF$ is represented by $f(x)$, and it must satisfy two properties: the function values are non-negative for all ‘$x$,’ and the total area under the curve is equal to 1.

Differences and Applications

The key distinction lies in the type of variable each function represents. PMF deals with discrete variables, such as the number of heads obtained in multiple coin tosses, or the outcomes of rolling a dice. PDF, on the other hand, is applicable to continuous variables like height measurements or time intervals.

Example: Consider a scenario where ‘$X$’ represents the number of defective items in a production batch. $X$ is a discrete random variable. The PMF of $X$ will provide the probabilities of obtaining 0, 1, 2, and so on, defective items in a batch.

Now, let’s consider a continuous random variable ‘$Y$’ representing the weight of a randomly selected apple from a basket. The PDF of ‘$Y$’ will specify the likelihood of the weight falling within a certain range, say between 150g and 200g.