Part 03: Describing Random Outcomes: PMF, CDF, and PDF

Author(s): Sudeep

Originally published on Towards AI.

In the previous article, we introduced the concept of a random experiment using the example of student marks in a class. Now, we will delve deeper into how we mathematically describe the likelihood of different outcomes using key functions from probability theory

To understand the results of a random experiment like observing student marks, we need ways to quantify the probabilities of different outcomes. This involves defining a random variable (like the score itself) and describing its probability distribution.

Let’s assume the “test score” (our random variable, let’s call it X) can only take discrete integer values from 1 to 100.

Probability Mass Function (PMF)

For a discrete random variable, the PMF gives the probability that the variable takes on exactly a specific value.

In simple terms: It answers, “What is the probability that the score X is exactly equal to x?” We write this as P(X = x).

Imagine a class of 10 students who took a mock test, and their scores (out of 100) are:
55, 70, 85, 60, 75, 90, 80, 95, 70, 65

Step 1: Frequency Calculation
Let’s calculate the frequency of each score:

• 55: 1 occurrence
• 60: 1 occurrence
• 65: 1 occurrence
• 70: 2 occurrences
• 75: 1 occurrence
• 80: 1 occurrence
• 85: 1 occurrence
• 90: 1 occurrence
• 95: 1 occurrence

Step 2: Calculate PMF
The PMF gives the probability of each mark occurring:

For example,

The PMF tells us the likelihood of students scoring exactly a specific mark. For instance, the probability of scoring 70 is 0.2 or 20%.

Cumulative Distribution Function (CDF)

The CDF gives the probability that a random variable (discrete or continuous) takes on a value less than or equal to a specific value x. It represents accumulated probability.

In simple terms: It answers, “What is the probability that the score X is at most x ?” We write this as F(x) = P(X ≤ x).

Example (Discrete Marks)

• Step 1: Cumulative Sum Calculation.

For example,

This means there’s a 50% chance a randomly chosen student scored 70 or less.

What if the Variable is Continuous? The Probability Density Function (PDF)

• Sometimes, variables can take any value within a range (e.g., height, exact time, or if marks could be 75. K5, 81.2, etc.). These are continuous random variables.
• For continuous variables, we use the Probability Density Function (PDF), denoted f(x). To find the probability over an interval, we calculate the area under the curve:

• Key Difference: The PDF f(x) itself does not give the probability that X equals x (that probability is actually 0 for continuous variables). Instead, the PDF describes the relative likelihood or density of the variable around the value x.
• CDF for Continuous: The CDF F(x) = P(X ≤ x) still works similarly, but it’s calculated by integrating the PDF from the minimum possible value up to x.

Real-World Applications

Education Analytics:

• PMF: Analyzing the frequency of specific scores to understand grade distributions.
• CDF: Determining the proportion of students scoring below a certain threshold.

Risk Assessment:

• Using PDFs and CDFs to evaluate the likelihood of certain financial events or insurance claims.

Quality Control:

• Estimating defect rates in manufacturing by modeling the probability of various outcomes.

Summary: Understanding Random Experiments

• Probability Mass Function (PMF): Describes the likelihood of discrete outcomes, answering “What is the probability that X equals x?”
• Cumulative Distribution Function (CDF): Accumulates probabilities up to a certain value, providing the chance that X is less than or equal to x.
• Probability Density Function (PDF): For continuous variables, describes the density of outcomes, where probabilities are calculated as areas under the curve.

Next stop: Estimation of Population and Hypothesis Testing.

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Published via Towards AI