Introduction

In the realm of probability theory and statistics, understanding the concepts of a Probability Mass Function (PMF) and a Cumulative Distribution Function (CDF) is fundamental. These functions play a crucial role in characterizing the behavior of random variables and their associated probabilities. This chapter delves into the definitions, distinctions, and the intricate relationship between the PMF and CDF, illuminating how these functions provide insights into the probabilities of specific outcomes and cumulative probabilities.

Probability Mass Function (PMF)

A Probability Mass Function (PMF) is a fundamental concept in probability theory, especially for discrete random variables. It defines the probability distribution of each possible value that a discrete random variable can take. In simpler terms, the PMF assigns probabilities to individual values within the variable’s domain. Formally, for a discrete random variable X, the PMF is defined as P(X = x), where x represents a specific value in the variable’s range.

Probability Mass Function (PMF)

A Probability Mass Function (PMF) is a fundamental concept in probability theory, especially for discrete random variables. It defines the probability distribution of each possible value that a discrete random variable can take. In simpler terms, the PMF assigns probabilities to individual values within the variable’s domain. Formally, for a discrete random variable $X$, the $P M F$ is defined as $P(X=x)$, where $x$ represents a specific value in the variable’s range.

Example 1.1: PDF for Coin Toss
Consider the random variable $X$ representing the number of heads when tossing a fair coin three times. The possible values of $X$ are $0,1,2$, and 3. The PMF of $X$ is:

• $P(X=0)=1 / 8$
• $P(X=1)=3 / 8$
• $P(X=2)=3 / 8$
• $P(X=3)=1 / 8$

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) provides a broader perspective by offering cumulative probabilities for values up to a certain point. For a discrete random variable, the CDF of $X$ is denoted by $F(x)$ and is defined as the sum of the probabilities of all values less than or equal to $x$.

Example: CDF for Coin Toss
Using the above example of the coin toss, the CDF $F(x)$ for $X$ is:

• $F(0)=P(X \leq 0)=1 / 8$
• $F(1)=P(X \leq 1)=4 / 8$
• $F(2)=P(X \leq 2)=7 / 8$
• $F(3)=P(X \leq 3)=1$

Relationship Between PMF and CDF

The connection between the PMF and CDF is intrinsic. The PMF provides the discrete probabilities of specific values, while the CDF accumulates these probabilities. Mathematically, for a discrete random variable $\mathrm{X}$ and a given value $\mathrm{x}$, the relationship can be represented as follows:

• PMF: $P(X=x)$
• CDF: $F(x)=P(X \leq x)$