12.8. PMF vs PDF vs CDF

Feature PMF PDF CDF Type of Discrete Continuous Both Variable

This chunk introduces the types of random variable functions: PMF, PDF, and CDF. The PMF (Probability Mass Function) is specifically for discrete random variables, meaning it is used when the possible outcomes can be counted (like the roll of a die). The PDF (Probability Density Function) is for continuous random variables; it describes the probabilities of outcomes in a range instead of distinct values (like the height of people). The CDF (Cumulative Distribution Function) can be used for both types, as it defines the probability that a random variable takes on a value less than or equal to a specific value.

Think of PMF as counting the number of apples in a basket (discrete), PDF as measuring the weight of apples on a scale (continuous), and CDF as keeping a running total of all of the apples counted so far (both).

Definition 𝑃(𝑋= 𝑥) 𝑓(𝑥) such that area under curve = 𝑃(𝑋 ≤ 𝑥)

Here, we define each function briefly. The PMF is defined as the probability that the discrete random variable X equals a specific value x (P(X = x)). The PDF is defined such that the area under the curve of the function equals the probability that the variable falls within a certain range. Meanwhile, the CDF is defined as the probability that the variable X is less than or equal to x. This means that you can use these definitions to calculate specific probabilities depending on whether your variables are discrete or continuous.

Imagine you're trying to find the probability of a die showing a '3' (PMF), the probability of rolling less than '4' (CDF), and the likelihood of rolling any number between '1 and 6' (PDF), where PDF would represent the continuous range of outcomes.

Graph Bars Smooth curve Step or continuous curve

This chunk describes how each function is visually represented. The PMF is represented by bars, where each bar's height corresponds to the probability of a specific outcome; the PDF is represented by a smooth curve, where the area under the curve represents probability; and the CDF can be either a step curve for discrete variables or a continuous curve for continuous variables. This visual distinction helps understand how data is distributed across different types.

Consider a bar-chart representing scores in a quiz (PMF), a smooth mountain shape showing the height distributions in a class (PDF), and a staircase-like graph showing the cumulative scores of students stepping up as they get good or poor scores (CDF).

Integration Not Used to derive CDF Not applicable

This chunk explains how these functions relate to probabilities. The PMF does not require integration since it directly gives the probability for discrete values. The PDF is used to derive the CDF through integration, meaning that we find the cumulative probability by calculating the area under the PDF curve. This is not applicable to PMF since it's based on counting probabilities. Thus, knowing when to use each function is crucial for calculating probabilities correctly.

Think of PMF as directly asking how many apples you have (counting them), PDF as pouring different amounts of water into a tank and measuring how much fills up (integration across the increments), and CDF as keeping track of the total amount of water that has been added so far.

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