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Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

CDF vs PDF: What’s the Difference?

Nitika Sharma
Nitika Sharma Last Updated : 01 Dec, 2024
7 min read

The Cumulative Distribution Function and the Probability Density Function are two essential ideas in probability theory that frequently confound students. Understanding random variables’ behavior, features, and distributions depends critically on these operations. Knowing the differences between PDF and CDF is crucial to analyzing and interpreting the probabilities linked to continuous and discrete random variables. This article will discuss the definitions of cumulative distribution function (CDF) vs probability density function (PDF) and their unique roles and interactions. We will also offer a solved example to show the difference between PDF and CDF use.

Overview:

  • Explore the differences between the Probability Density Function (PDF) and Cumulative Distribution Function (CDF), essential for understanding random variables.
  • Learn how PDF vs CDF provides complementary perspectives in probability theory—PDF shows probability density, and CDF illustrates cumulative probability.
  • Explore how CDF vs PDF helps interpret the behaviour and distribution of continuous and discrete random variables with practical examples.
  • Understand the mathematical relationship between probability density function vs cumulative distribution function, highlighting how the CDF is derived from the PDF.
  • Dive into real-world PDF and CDF use cases, including statistical modeling, distribution analysis, and probability estimations.

Table of contents

What is the Probability Density Function (PDF)?

The PDF is a crucial tool for understanding the probabilities associated with continuous random variables. It provides a smooth curve representing the probability distribution over possible values. The PDF function does not give the probabilities of specific individual values. Still, it describes the likelihood of the random variable taking on values within a small interval around a particular point.

To understand the concept of PDF, imagine a continuous probability distribution, such as the height of adult males. The probability for various height ranges will be displayed in the PDF. It might suggest, for instance, that people with heights between 5’9″ and 5’10” are more numerous than those with heights outside of that range.

The area under the PDF curve spanning a range represents the probability that the random variable will fall inside that range. To calculate the probability of a single value, which is the probability that the random variable will be infinitesimally close to that value, you must compute the integral of the PDF at that point.

When comparing cumulative distribution function vs probability density function, it’s essential to understand their distinct purposes and applications.

What is Probability Density Function (PDF)?
Source: ResearchGate

What is Cumulative Distribution Function (CDF)?

The CDF is a complementary concept to the PDF and provides a cumulative perspective of the probabilities associated with a random variable. Unlike the smooth curve of the PDF, the CDF is a step function that jumps at specific values. It displays the likelihood that a particular number will be less than or equal to the random variable.

The CDF begins at 0 for negative values, moving steadily towards 1 as the random variable’s value rises. For discrete random variables, the CDF increases in steps corresponding to the probabilities of each possible outcome. For continuous random variables, it increases smoothly, reflecting the accumulated probabilities across different intervals.

The CDF would demonstrate the likelihood of discovering a male with a height less than or equal to a certain value, such as 5 ‘9″, using the male heights example from before. By presenting cumulative probability, the CDF allows us to respond to questions like “What percentage of adult males is shorter than 5 ‘9”?

What is Cumulative Distribution Function (CDF)?

Also Read: 6 Types of Probability Distribution in Data Science

PDF vs CDF Understanding with Example

Understanding how the Probability Density Function (PDF) vs Cumulative Distribution Function (CDF) interact is essential for comprehending how random variables behave and how their distributions work. Both functions provide complementary insights into the probabilities of the random variable’s values.

We previously showed how to compute the PDF vs CDF using the fair six-sided die example. Let’s now explore their connection and deeper aspects of their relationship.

Also, Read this for more information click here

Calculating the CDF from the PDF

We need to integrate the PDF over a given range to find the CDF from the PDF. The CDF at a certain point x (F(x)) for a continuous random variable equals the region of the PDF curve up to that point. It can be modelled mathematically as follows:

F(x)=[a, x]f(t)dt

Here, x is the point on the distribution curve for which we wish to get the cumulative probability, and an is the lower limit of the range.

For our example of rolling the fair die, we can use the PDF values we previously calculated to find the CDF:

Let's calculate the CDF at x = 3:

F(3) = ∫[1, 3] f(t) dt

F(3) = ∫[1, 3] 16 dt

F(3) = [t6] |[1, 3]

F(3) = (36) - (16)

F(3) = 26

Similarly, we can calculate the CDF for other values of x using the same approach.

Relating PDF and CDF for Discrete Random Variables

The relationship between the PMF (Probability Mass Function) and the CDF is more apparent for discrete random variables. The PMF provides the probabilities for each specific value of the discrete random variable, while the CDF accumulates these probabilities.

The CDF at a particular value, x, is the sum of all the probabilities of the random variable being less than or equal to x. Mathematically, for discrete random variables:

F(x) = P(X ≤ x) = Σ[all values ≤ x] P(X = value)

By adding up the probabilities of all values up to x, we obtain the cumulative probability up to that point, which aligns with the CDF concept.

Checkout: 40 Questions on Probability for Data Science Professionals

Understanding the Difference Between CDF vs PDF

Let us now understand the difference between PDF and CDF.

The CDF provides the probability that a random variable is less than or equal to a specific value, ‘x.’ The PDF represents the probability that the random variable takes on a precise value, ‘x.’

Let’s understand the unique properties and applications in PDF and CDP:

Understanding Difference Between CDF vs PDF
Source: Haslwanter

CDF vs PDF: Definition

PDFCDF
The probability density function or PDF describes a continuous random variable’s probability distribution. It shows the probability that the random variable will have a particular value.In general, the probability that a random variable will have a value less than or equal to a specific value is determined by the cumulative distribution function or CDF.

CDF vs PDF: Representation

PDFCDF
A continuous random variable is frequently represented using the expression f(x), where ‘x’ represents the variable’s value.It can be applied to continuous and discrete random variables and is frequently expressed as F(x), where ‘x’ represents the variable’s value.

CDF vs PDF: Function Type

PDFCDF
The PDF is used for continuous random variables, where the probability is distributed over an infinite range of values.The CDF applies to discrete and continuous random variables, as it accumulates probabilities for all possible values of the random variable.

CDF vs PDF: Interpretation

PDFCDF
The PDF provides the probability density at a particular point on the continuous distribution curve, indicating how the probability is spread across different values.The CDF gives the cumulative probability up to a specific value, offering insights into the probabilities of the random variable being less than or equal to that value.

CDF vs PDF: Integration

PDFCDF
The integral of the PDF over a certain range yields the probability of the random variable falling within that range.The CDF is obtained by integrating the PDF from a lower bound to a specific value, ‘x’, which accumulates the probabilities up to that point.

CDF vs PDF: Range

PDFCDF
The PDF can take any non-negative value for any given point on the distribution curve, representing the likelihood of the variable assuming that value.The CDF always ranges from 0 to 1, as it gives the cumulative probability, and it is non-decreasing, meaning it can only increase or remain constant as ‘x’ increases.

CDF vs PDF: Application

PDFCDF
The PDF is commonly used in probability density estimation, statistical modelling, and understanding the shape of continuous distributions.The CDF can be used to determine a distribution’s percentiles and quantiles and the likelihood that a random variable will fall within a certain range.

Conclusion

Understanding PDF and CDF differences is crucial for interpreting random variables’ distributions and behaviors in probability theory. The PDF and CDF serve distinct yet complementary roles: while the PDF provides the probability density of continuous random variables, showing the likelihood of values within specific intervals, the CDF accumulates probabilities, illustrating the likelihood of a variable being less than or equal to a particular value. Comparing the cumulative distribution function vs probability density function helps in appreciating their unique contributions to probability theory.

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Frequently Asked Questions

Q1. What is the relationship between PDF and CDF?

A. The PDF and CDF are interrelated concepts in probability theory. The PDF gives the probability of a continuous random variable taking on a specific value. At the same time, the CDF provides the cumulative probability of the random variable being less than or equal to a given value.

Q2. What are CDF and PDF functions?

A. The CDF and PDF are important in probability and statistics for describing random variable behavior. The CDF shows the cumulative probability up to a specific value “x” (denoted as “F(x)”). At the same time, the PDF displays the probability distribution of a continuous random variable (represented as “f(x)”).

Q3. What is the difference between PDF and PMF?

A. PMF is for discrete random variables, giving probabilities for specific values. On the other hand, the PDF is for continuous random variables, showing the probability density over a range of values.

Q4. What is the difference between the probability distribution function and the probability density function?

A. Both terms represent a mathematical function describing the probability distribution of a continuous random variable. Though “probability density function” and “probability distribution function” are interchangeable, they mean the same thing.

Q5. Is CDF just the integral of PDF?

A. Yes, CDF is the integral of PDF for continuous variables. Think of it like this:
PDF: How likely a specific value is.
CDF: How likely a value less than or equal to that specific value is.
The CDF builds up the probability by integrating the PDF.

Nitika Sharma
Nitika Sharma

Hello, I am Nitika, a tech-savvy Content Creator and Marketer. Creativity and learning new things come naturally to me. I have expertise in creating result-driven content strategies. I am well versed in SEO Management, Keyword Operations, Web Content Writing, Communication, Content Strategy, Editing, and Writing.

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