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

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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 be 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.

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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

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