Back To Course

Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Artem Cheprasov*

Continuous probability distributions can be a good approximation of many real world processes and phenomena. In this lesson, you will gain a conceptual understanding of continuous probability distributions and how to apply their properties to solve problems.

Have you ever competed in a marathon? Many sports competitions involve measuring the time to complete a certain task, whether it would be running, swimming, kayaking, and so on down the line. Although the time measurements used to rank the participants are rounded to a certain decimal value, in principle, there is an infinite precision associated with such measurements. For example, we could measure the time to be 37.25 seconds or 37.248 seconds or 37.247899 seconds.

In this lesson, you will learn about how to apply continuous probability concepts to solve problems. In particular, we will examine how to find areas under the curve of continuous probability distributions.

There are two types of probability distributions: continuous and discrete.

A discrete probability distribution is associated with processes such as flipping a coin and rolling dice. In this case, there is a countable number of possible outcomes. For example, the possible outcomes of a coin flip are heads and tails, while the possible outcomes of rolling a six-sided die are integers from 1 to 6, inclusive.

A **continuous probability distribution** is a model of processes in which there is an uncountable number of possible outcomes. Weight and height measurements within a population would be associated with a continuous probability distribution. This is because there is no mathematical limit, at least in theory, on the precision with which these quantities can be measured.

Let's proceed to discuss some continuous probability concepts that are useful for problem solving. The graph of a continuous probability distribution is, as you might guess, continuous. This means that it does not contain any holes, jumps, or vertical asymptotes. In addition, a continuous probability distribution function, *f(x)*, also referred to as the **probability density function**, must satisfy the properties shown on the screen (see video).

1. This first property implies that there could not be a negative probability for a certain outcome. In other words, the lowest possible probability associated with any outcome is zero.

2. This second property indicates that the total area under the curve of the probability density function must equal to one.

3. The third property implies that the probability for a range of values can be found by calculating the area under the curve of the probability density function.

How can these properties be applied to problems? Let's work out some examples together.

The probability density function associated with the resistance of an electronic component is given by the function shown.

What is the probability that the resistance is less than 3?

Since the probability is non-zero only within the range of 1 to 4, the limits on the integral are from 1 to 3. In other words, we compute the area under the curve of the probability density function to the left of 3. The solution to this problem is as shown.

Next, what is the total area under the curve of the probability density function? To solve this problem, we simply take the integral over the range in which the probability density function is non-zero, which happens to be between 1 and 4. Computing the integral gives us 1 for an answer, as we should expect. The calculation is shown.

And finally, is the probability always positive? Looking at how the function is defined, there is no *x* for which *f(x)* is less than zero. This is in agreement with what we should expect from the probability density function.

This may have been a dense lesson causing your brain to malfunction. So, let's summarize everything.

In this lesson, we have learned how to apply continuous probability concepts to solve problems. Recall that a continuous function does not contain any jumps, holes, or vertical asymptotes. Applying this concept to probabilities, a **continuous probability distribution** is a model of processes in which there is an uncountable number of possible outcomes. This is in contrast to a discrete probability distribution, which is associated with processes such as rolling dice and flipping a coin.

We have also learned that the **probability density function**, *f(x)*, must satisfy the following properties:

*f(x)*is always greater than or equal to zero- When
*f(x)*is integrated over its entire domain, the area under the curve is equal to one, and - The probability for a range of values can be found by calculating the area under the curve of the probability density function

When solving problems involving a continuous probability distribution, it's important to think about these properties and apply them properly. You should now be more comfortable in solving problems similar to the example we have done.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
11 in chapter 6 of the course:

Back To Course

Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

- Go to Probability

- Graphing Probability Distributions Associated with Random Variables 6:33
- Finding & Interpreting the Expected Value of a Continuous Random Variable 5:29
- Developing Continuous Probability Distributions Theoretically & Finding Expected Values 6:12
- Probabilities as Areas of Geometric Regions: Definition & Examples 7:06
- Normal Distribution: Definition, Properties, Characteristics & Example 11:40
- Finding Z-Scores: Definition & Examples 6:30
- Estimating Areas Under the Normal Curve Using Z-Scores 5:54
- Estimating Population Percentages from Normal Distributions: The Empirical Rule & Examples 4:41
- Using the Normal Distribution: Practice Problems 10:32
- Using Normal Distribution to Approximate Binomial Probabilities 6:34
- How to Apply Continuous Probability Concepts to Problem Solving 5:05
- Go to Continuous Probability Distributions

- Go to Sampling

- GRE Information Guide
- Computer Science 310: Current Trends in Computer Science & IT
- Earth Science 105: Introduction to Oceanography
- Computer Science 331: Cybersecurity Risk Analysis Management
- Computer Science 336: Network Forensics
- World Literature: Drama Since the 20th Century
- Visual Art Since the 18th Century
- World Literature: Drama Through the 19th Century
- Defamation, Libel & Slander
- Elements of Music Overview
- ILTS Prep Product Comparison
- CTEL Prep Product Comparison
- TASC Prep Product Comparison
- FSA Prep Product Comparison
- SHSAT Prep Product Comparison
- MEGA Test Accomodations
- Study.com Grant for Teachers

- Materials & Resources for an Early Childhood Classroom
- Obstructive Shock: Causes, Symptoms & Treatment
- Interpreting & Calculating Seasonal Indices
- Managing Classroom Behaviors of Young Children
- Taekwondo Lesson Plan
- Normalization & Invisibility of Privilege in the Workplace
- Practical Application: Reducing Job Stress Using Time Management
- Solving Equations Using the Least Common Multiple
- Quiz & Worksheet - Real-World Applications of Learning
- Quiz & Worksheet - Dante's Inferno 4th Level of Hell
- Quiz & Worksheet - Coaching Agreements
- Quiz & Worksheet - Code of Ethics for Teaching
- Quiz & Worksheet - Third-Person Pronouns
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Civil War History: Homework Help
- Business 116: Quantitative Analysis
- UExcel Introduction to Macroeconomics: Study Guide & Test Prep
- The Vietnam War: Help and Review
- Praxis Psychology (5391): Practice & Study Guide
- AEPA Math: Vector Operations
- Unit Rates & Ratios of Fractions: CCSS.Math.Content.7.RP.A.1
- Quiz & Worksheet - What is Conversational Disclosure?
- Quiz & Worksheet - Number Systems & the Base-Ten System
- Quiz & Worksheet - US Citizenship Process & Citizens' Duties
- Quiz & Worksheet - Answering Mirror Questions With Equations
- Quiz & Worksheet - Factors Impacting Developmental Genes

- Cell Fate Specification: Cytoplasmic Determinants & Inductive Signals
- Carol Dweck & Growth Mindset Psychology
- How to Earn a Micro Degree
- Easter Bulletin Board Ideas
- Community College Teaching Jobs
- Social Studies Games for Kids
- Math Project Rubrics
- Fun Math Games for 4th Grade
- STAAR Test Taking Strategies
- Ohio Alternative Teacher Certification
- How Long Does it Take to Learn Spanish?
- Course Curriculum Template

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject