- Blog Categories
- Software Development Projects and Ideas
- 12 Computer Science Project Ideas
- 28 Beginner Software Projects
- Top 10 Engineering Project Ideas
- Top 10 Easy Final Year Projects
- Top 10 Mini Projects for Engineers
- 25 Best Django Project Ideas
- Top 20 MERN Stack Project Ideas
- Top 12 Real Time Projects
- Top 6 Major CSE Projects
- 12 Robotics Projects for All Levels
- Java Programming Concepts
- Abstract Class in Java and Methods
- Constructor Overloading in Java
- StringBuffer vs StringBuilder
- Java Identifiers: Syntax & Examples
- Types of Variables in Java Explained
- Composition in Java: Examples
- Append in Java: Implementation
- Loose Coupling vs Tight Coupling
- Integrity Constraints in DBMS
- Different Types of Operators Explained
- Career and Interview Preparation in IT
- Top 14 IT Courses for Jobs
- Top 20 Highest Paying Languages
- 23 Top CS Interview Q&A
- Best IT Jobs without Coding
- Software Engineer Salary in India
- 44 Agile Methodology Interview Q&A
- 10 Software Engineering Challenges
- Top 15 Tech's Daily Life Impact
- 10 Best Backends for React
- Cloud Computing Reference Models
- Web Development and Security
- Find Installed NPM Version
- Install Specific NPM Package Version
- Make API Calls in Angular
- Install Bootstrap in Angular
- Use Axios in React: Guide
- StrictMode in React: Usage
- 75 Cyber Security Research Topics
- Top 7 Languages for Ethical Hacking
- Top 20 Docker Commands
- Advantages of OOP
- Data Science Projects and Applications
- 42 Python Project Ideas for Beginners
- 13 Data Science Project Ideas
- 13 Data Structure Project Ideas
- 12 Real-World Python Applications
- Python Banking Project
- Data Science Course Eligibility
- Association Rule Mining Overview
- Cluster Analysis in Data Mining
- Classification in Data Mining
- KDD Process in Data Mining
- Data Structures and Algorithms
- Binary Tree Types Explained
- Binary Search Algorithm
- Sorting in Data Structure
- Binary Tree in Data Structure
- Binary Tree vs Binary Search Tree
- Recursion in Data Structure
- Data Structure Search Methods: Explained
- Binary Tree Interview Q&A
- Linear vs Binary Search
- Priority Queue Overview
- Python Programming and Tools
- Top 30 Python Pattern Programs
- List vs Tuple
- Python Free Online Course
- Method Overriding in Python
- Top 21 Python Developer Skills
- Reverse a Number in Python
- Switch Case Functions in Python
- Info Retrieval System Overview
- Reverse a Number in Python
- Real-World Python Applications
- Data Science Careers and Comparisons
- Data Analyst Salary in India
- Data Scientist Salary in India
- Free Excel Certification Course
- Actuary Salary in India
- Data Analyst Interview Guide
- Pandas Interview Guide
- Tableau Filters Explained
- Data Mining Techniques Overview
- Data Analytics Lifecycle Phases
- Data Science Vs Analytics Comparison
- Artificial Intelligence and Machine Learning Projects
- Exciting IoT Project Ideas
- 16 Exciting AI Project Ideas
- 45+ Interesting ML Project Ideas
- Exciting Deep Learning Projects
- 12 Intriguing Linear Regression Projects
- 13 Neural Network Projects
- 5 Exciting Image Processing Projects
- Top 8 Thrilling AWS Projects
- 12 Engaging AI Projects in Python
- NLP Projects for Beginners
- Concepts and Algorithms in AIML
- Basic CNN Architecture Explained
- 6 Types of Regression Models
- Data Preprocessing Steps
- Bagging vs Boosting in ML
- Multinomial Naive Bayes Overview
- Gini Index for Decision Trees
- Bayesian Network Example
- Bayes Theorem Guide
- Top 10 Dimensionality Reduction Techniques
- Neural Network Step-by-Step Guide
- Technical Guides and Comparisons
- Make a Chatbot in Python
- Compute Square Roots in Python
- Permutation vs Combination
- Image Segmentation Techniques
- Generative AI vs Traditional AI
- AI vs Human Intelligence
- Random Forest vs Decision Tree
- Neural Network Overview
- Perceptron Learning Algorithm
- Selection Sort Algorithm
- Career and Practical Applications in AIML
- AI Salary in India Overview
- Biological Neural Network Basics
- Top 10 AI Challenges
- Production System in AI
- Top 8 Raspberry Pi Alternatives
- Top 8 Open Source Projects
- 14 Raspberry Pi Project Ideas
- 15 MATLAB Project Ideas
- Top 10 Python NLP Libraries
- Naive Bayes Explained
- Digital Marketing Projects and Strategies
- 10 Best Digital Marketing Projects
- 17 Fun Social Media Projects
- Top 6 SEO Project Ideas
- Digital Marketing Case Studies
- Coca-Cola Marketing Strategy
- Nestle Marketing Strategy Analysis
- Zomato Marketing Strategy
- Monetize Instagram Guide
- Become a Successful Instagram Influencer
- 8 Best Lead Generation Techniques
- Digital Marketing Careers and Salaries
- Digital Marketing Salary in India
- Top 10 Highest Paying Marketing Jobs
- Highest Paying Digital Marketing Jobs
- SEO Salary in India
- Brand Manager Salary in India
- Content Writer Salary Guide
- Digital Marketing Executive Roles
- Career in Digital Marketing Guide
- Future of Digital Marketing
- MBA in Digital Marketing Overview
- Digital Marketing Techniques and Channels
- 9 Types of Digital Marketing Channels
- Top 10 Benefits of Marketing Branding
- 100 Best YouTube Channel Ideas
- YouTube Earnings in India
- 7 Reasons to Study Digital Marketing
- Top 10 Digital Marketing Objectives
- 10 Best Digital Marketing Blogs
- Top 5 Industries Using Digital Marketing
- Growth of Digital Marketing in India
- Top Career Options in Marketing
- Interview Preparation and Skills
- 73 Google Analytics Interview Q&A
- 56 Social Media Marketing Q&A
- 78 Google AdWords Interview Q&A
- Top 133 SEO Interview Q&A
- 27+ Digital Marketing Q&A
- Digital Marketing Free Course
- Top 9 Skills for PPC Analysts
- Movies with Successful Social Media Campaigns
- Marketing Communication Steps
- Top 10 Reasons to Be an Affiliate Marketer
- Career Options and Paths
- Top 25 Highest Paying Jobs India
- Top 25 Highest Paying Jobs World
- Top 10 Highest Paid Commerce Job
- Career Options After 12th Arts
- Top 7 Commerce Courses Without Maths
- Top 7 Career Options After PCB
- Best Career Options for Commerce
- Career Options After 12th CS
- Top 10 Career Options After 10th
- 8 Best Career Options After BA
- Projects and Academic Pursuits
- 17 Exciting Final Year Projects
- Top 12 Commerce Project Topics
- Top 13 BCA Project Ideas
- Career Options After 12th Science
- Top 15 CS Jobs in India
- 12 Best Career Options After M.Com
- 9 Best Career Options After B.Sc
- 7 Best Career Options After BCA
- 22 Best Career Options After MCA
- 16 Top Career Options After CE
- Courses and Certifications
- 10 Best Job-Oriented Courses
- Best Online Computer Courses
- Top 15 Trending Online Courses
- Top 19 High Salary Certificate Courses
- 21 Best Programming Courses for Jobs
- What is SGPA? Convert to CGPA
- GPA to Percentage Calculator
- Highest Salary Engineering Stream
- 15 Top Career Options After Engineering
- 6 Top Career Options After BBA
- Job Market and Interview Preparation
- Why Should You Be Hired: 5 Answers
- Top 10 Future Career Options
- Top 15 Highest Paid IT Jobs India
- 5 Common Guesstimate Interview Q&A
- Average CEO Salary: Top Paid CEOs
- Career Options in Political Science
- Top 15 Highest Paying Non-IT Jobs
- Cover Letter Examples for Jobs
- Top 5 Highest Paying Freelance Jobs
- Top 10 Highest Paying Companies India
- Career Options and Paths After MBA
- 20 Best Careers After B.Com
- Career Options After MBA Marketing
- Top 14 Careers After MBA In HR
- Top 10 Highest Paying HR Jobs India
- How to Become an Investment Banker
- Career Options After MBA - High Paying
- Scope of MBA in Operations Management
- Best MBA for Working Professionals India
- MBA After BA - Is It Right For You?
- Best Online MBA Courses India
- MBA Project Ideas and Topics
- 11 Exciting MBA HR Project Ideas
- Top 15 MBA Project Ideas
- 18 Exciting MBA Marketing Projects
- MBA Project Ideas: Consumer Behavior
- What is Brand Management?
- What is Holistic Marketing?
- What is Green Marketing?
- Intro to Organizational Behavior Model
- Tech Skills Every MBA Should Learn
- Most Demanding Short Term Courses MBA
- MBA Salary, Resume, and Skills
- MBA Salary in India
- HR Salary in India
- Investment Banker Salary India
- MBA Resume Samples
- Sample SOP for MBA
- Sample SOP for Internship
- 7 Ways MBA Helps Your Career
- Must-have Skills in Sales Career
- 8 Skills MBA Helps You Improve
- Top 20+ SAP FICO Interview Q&A
- MBA Specializations and Comparative Guides
- Why MBA After B.Tech? 5 Reasons
- How to Answer 'Why MBA After Engineering?'
- Why MBA in Finance
- MBA After BSc: 10 Reasons
- Which MBA Specialization to choose?
- Top 10 MBA Specializations
- MBA vs Masters: Which to Choose?
- Benefits of MBA After CA
- 5 Steps to Management Consultant
- 37 Must-Read HR Interview Q&A
- Fundamentals and Theories of Management
- What is Management? Objectives & Functions
- Nature and Scope of Management
- Decision Making in Management
- Management Process: Definition & Functions
- Importance of Management
- What are Motivation Theories?
- Tools of Financial Statement Analysis
- Negotiation Skills: Definition & Benefits
- Career Development in HRM
- Top 20 Must-Have HRM Policies
- Project and Supply Chain Management
- Top 20 Project Management Case Studies
- 10 Innovative Supply Chain Projects
- Latest Management Project Topics
- 10 Project Management Project Ideas
- 6 Types of Supply Chain Models
- Top 10 Advantages of SCM
- Top 10 Supply Chain Books
- What is Project Description?
- Top 10 Project Management Companies
- Best Project Management Courses Online
- Salaries and Career Paths in Management
- Project Manager Salary in India
- Average Product Manager Salary India
- Supply Chain Management Salary India
- Salary After BBA in India
- PGDM Salary in India
- Top 7 Career Options in Management
- CSPO Certification Cost
- Why Choose Product Management?
- Product Management in Pharma
- Product Design in Operations Management
- Industry-Specific Management and Case Studies
- Amazon Business Case Study
- Service Delivery Manager Job
- Product Management Examples
- Product Management in Automobiles
- Product Management in Banking
- Sample SOP for Business Management
- Video Game Design Components
- Top 5 Business Courses India
- Free Management Online Course
- SCM Interview Q&A
- Fundamentals and Types of Law
- Acceptance in Contract Law
- Offer in Contract Law
- 9 Types of Evidence
- Types of Law in India
- Introduction to Contract Law
- Negotiable Instrument Act
- Corporate Tax Basics
- Intellectual Property Law
- Workmen Compensation Explained
- Lawyer vs Advocate Difference
- Law Education and Courses
- LLM Subjects & Syllabus
- Corporate Law Subjects
- LLM Course Duration
- Top 10 Online LLM Courses
- Online LLM Degree
- Step-by-Step Guide to Studying Law
- Top 5 Law Books to Read
- Why Legal Studies?
- Pursuing a Career in Law
- How to Become Lawyer in India
- Career Options and Salaries in Law
- Career Options in Law India
- Corporate Lawyer Salary India
- How To Become a Corporate Lawyer
- Career in Law: Starting, Salary
- Career Opportunities: Corporate Law
- Business Lawyer: Role & Salary Info
- Average Lawyer Salary India
- Top Career Options for Lawyers
- Types of Lawyers in India
- Steps to Become SC Lawyer in India
- Tutorials
- C Tutorials
- Recursion in C: Fibonacci Series
- Checking String Palindromes in C
- Prime Number Program in C
- Implementing Square Root in C
- Matrix Multiplication in C
- Understanding Double Data Type
- Factorial of a Number in C
- Structure of a C Program
- Building a Calculator Program in C
- Compiling C Programs on Linux
- Java Tutorials
- Handling String Input in Java
- Determining Even and Odd Numbers
- Prime Number Checker
- Sorting a String
- User-Defined Exceptions
- Understanding the Thread Life Cycle
- Swapping Two Numbers
- Using Final Classes
- Area of a Triangle
- Skills
- Software Engineering
- JavaScript
- Data Structure
- React.js
- Core Java
- Node.js
- Blockchain
- SQL
- Full stack development
- Devops
- NFT
- BigData
- Cyber Security
- Cloud Computing
- Database Design with MySQL
- Cryptocurrency
- Python
- Digital Marketings
- Advertising
- Influencer Marketing
- Search Engine Optimization
- Performance Marketing
- Search Engine Marketing
- Email Marketing
- Content Marketing
- Social Media Marketing
- Display Advertising
- Marketing Analytics
- Web Analytics
- Affiliate Marketing
- MBA
- MBA in Finance
- MBA in HR
- MBA in Marketing
- MBA in Business Analytics
- MBA in Operations Management
- MBA in International Business
- MBA in Information Technology
- MBA in Healthcare Management
- MBA In General Management
- MBA in Agriculture
- MBA in Supply Chain Management
- MBA in Entrepreneurship
- MBA in Project Management
- Management Program
- Consumer Behaviour
- Supply Chain Management
- Financial Analytics
- Introduction to Fintech
- Introduction to HR Analytics
- Fundamentals of Communication
- Art of Effective Communication
- Introduction to Research Methodology
- Mastering Sales Technique
- Business Communication
- Fundamentals of Journalism
- Economics Masterclass
- Free Courses
Introduction to Probability Density Function [Formula, Properties, Applications, Examples]
Updated on 27 September, 2022
9.74K+ views
• 9 min read
Table of Contents
- Conditions to be Satisfied by a function to be considered a Probability Density Function
- Difference between Probability Density Function and Probability Distribution Function
- Expression for Probability Density Functions
- The formula of Probability Density Function
- Properties of a Probability Density Function
- Properties of a Probability Density Function
- Applications of Probability Density Function
- Examples of Probability Density Function
- Conclusion
Probability Density Function (PDF) is an expression in statistics that denotes the probability distribution of a discrete random variable. Probability distribution, in simple terms, can be defined as a likelihood of an outcome of a random variable like a stock or an ETF. Discrete variables occur in contrast to a continuous random variable whose accurate value can be determined.
Best Machine Learning and AI Courses Online
For instance, the value of scrip in a stock market has only two decimal points (for example, 65.76) in a discrete random variable instead of a continuous variable with any number of decimal points (example: 65.7685434567).
A probability density function is a statistical tool used to determine the likelihood of the outcome of a discrete random variable. When plotted on a graph, PDFs look identical to a bell curve in which the area under the curve represents the probability of the outcome.
In-demand Machine Learning Skills
Get Machine Learning Certification from the World’s top Universities. Earn Masters, Executive PGP, or Advanced Certificate Programs to fast-track your career.
When projected as a graphic model, the area under the curve represents the range in which the values of the discrete random variables will fall. Thus, the total area under the curve is equal to the probability of the variable’s outcome.
The probability density function can determine the likelihood of a random variable falling within a specific range of values.
Typically, probability density functions analyse the risks and potential revenue associated with a specific fund in the stock market.
Conditions to be Satisfied by a function to be considered a Probability Density Function
The value of a discrete variable can be accurately measured in contrast to a continuous variable that can have an infinite number of values. Any function should satisfy the below two conditions to be a probability density function:
- The f(x) value for each possible value of the random variable should be positive (non-negative).
- The integral value of the total area of the curve (integral of all possible values of the random variable) should be 1.
Difference between Probability Density Function and Probability Distribution Function
Random variables can have many values. The description of each possible value that a random variable can have is called its probability distribution.
The probability distribution gives a set of outcomes and their related probabilities. The statistical function that represents a continuous probability distribution is known as the probability density function.
There is another statistical tool that represents a discrete probability distribution called the probability mass function. This gives a detailed account of all the possible outcomes and their likelihood probabilities.
Expression for Probability Density Functions
If the random variable is discrete, its probability distribution is called probability mass function, and if it is a continuous variable, the probability distribution is called probability density function.
A PDF is used when the random variable in question has a range of possible values. Their probability distribution is used to determine the exact value.
Let the random variable be denoted by X. The probability density function, f of the random variable X can be expressed as
- The value of the random variable lies between a and b.
- If X denotes the probability of selecting a particular number from the range (interval) r and s, then the probability density function can be expressed as
f(x) = 1/(s − r) for r < x < s and f(x) = 0 for x < r or x > s.
- The PDF F is represented as:
F(x) = P{X ≤ x}
which is called the distribution function or the cumulative distribution function of X.
Considering the random variable X has a probability distribution function f(x), then the relationship between f and F can be established as
F′(.x) = f(x)
The distribution function of a discrete random variable is different from its probability distribution function. The relationship between the two can be expressed as below:
The expectation of the random variable is denoted as,
Thus, all discrete and random variables can be treated uniformly with the help of a combined theory.
The formula of Probability Density Function
The probability of a continuous random variable X on some fixed value x is always 0. In this case, P(X = x) cannot be used. The value of the X lying between a range of values (a,b) should be determined. To determine the same, the following formula is used.
Properties of a Probability Density Function
A continuous random variable that takes its value between the range (a,b), for instance, will be estimated by calculating the area under the curve and the X-axis plotted with (a) as its lower limit and (b) as its upper limit. The probability density function for the above is represented as:
Properties of a Probability Density Function
A continuous random variable that takes its value between the range (a,b), for instance, will be estimated by calculating the area under the curve and the X-axis plotted with (a) as its lower limit and (b) as its upper limit. The probability density function for the above is represented as:
The probability density function is positive (non-negative) for all possible values. This means f(x)≥ 0, for every x. The area falling between the density curve and the X-axis (horizontal axis) equals 1.
This can also be denoted as:
The density function curve is continuous throughout the given range, which is clearly defined against a series of continuous values or the variable’s domain.
Applications of Probability Density Function
- The probability density function is used in the yearly modelling of atmospheric NO concentration levels.
- Modelling of diesel engine combustion.
- In statistics, the probability density function is used to determine the possibilities of the outcome of a random variable.
Examples of Probability Density Function
Example 1
Below is an example of how probability density function (PDF) is used to determine the risk potential of an investor in the stock market:
First, PDFs are generated as a graphic tool based on historical information.
The most common form of PDF is the neutral projection, where the risk is equal to the reward across a range of possibilities. Investors with less risk-taking capability will only be rewarded with limited profits, and hence they come under the left side of the bell curve. Conversely, investors with high risk-taking abilities are likely to be rewarded with higher yields, and therefore, fall under the right side of the curve.
Most of the investors fall under average risk-taking ability, and hence they occupy the middle of the curve.
This helps in analysing the category of investors based on the data received. This helps stock market brokers to identify their target category of customers to sell their products.
Example 2
One of the essential applications of the probability density function is the Gaussian random variable, also known as a normal random variable.
In both cases, the graph gives a bell curve for the probability density function.
The density can be expressed as
The graph of the above density equation is given below.
The area under the curve represents the actual value of the Gaussian random variable.
Popular AI and ML Blogs & Free Courses
Conclusion
The probability density function plays a vital role in machine learning. For students eyeing a career in Machine Learning and Artificial Intelligence, we highly recommend enrolling in upGrad’s IIIT-Bangalore Master of Science in Machine Learning & AI. The program is customized and designed to equip senior working professionals to deploy machine learning models using cloud computing technology.
The curriculum is designed by faculty from IIT Madras and industry experts to make the learning process more relevant and practical. The program offers globally recognized certification from the coveted and No. 1 ranked Engineering college in India and 360-degree placement support from upGrad.
What’s more, you get numerous opportunities to collaborate on large-scale projects with upGrad’s paid learner base of 40,000+.
Head over to our website to begin your learning journey!
Frequently Asked Questions (FAQs)
1. Can a Probability Density Function be greater than 1?
As the probability function gives a fixed probability, it cannot be more than 1. A PDF f(x), however, can have values greater than 1 for certain values of X. This can happen as they represent the probable values (range for the area under the curve) and not the exact values of f(x).
2. What can be inferred from the probability density function?
The probability density function is the statistical technique used to determine the possibility of the outcome of a discrete random variable. The PDFs are depicted on a graph with the background data plotted in X and Y axes. The graph gives out a bell curve. The range of the curve gives us the range of the possible values, and the area under the curve provides the exact value of the discrete random variable.
3. What will be the probability density function of normal distribution?
A normal distribution is symmetric and has a non-zero probability for all positive and negative values of the random variable. The non-zero probability holds good even if the probability is assigned to values with more than 3 or 4 standard deviations as the mean is negligible.
RELATED PROGRAMS