Use The Uniform Probability Distribution To Solve The Following Question:What Is The Area Of The Part Of The Region Where 1.5 ≤ X ≤ 3 1.5 \leq X \leq 3 1.5 ≤ X ≤ 3 ?A. 1 6 \frac{1}{6} 6 1 ​ B. 1 3 \frac{1}{3} 3 1 ​ C. 1 2 \frac{1}{2} 2 1 ​ D. 2 3 \frac{2}{3} 3 2 ​

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Introduction

In probability theory, the uniform distribution is a continuous probability distribution where every possible outcome has an equal likelihood of occurring. This distribution is often used to model situations where all possible outcomes are equally likely. In this article, we will use the uniform probability distribution to solve a problem involving the area of a region.

Understanding the Uniform Distribution

The uniform distribution is a continuous probability distribution with a probability density function (PDF) given by:

f(x) = 1 / (b - a)

where a and b are the lower and upper bounds of the distribution, respectively. The uniform distribution has a constant probability density over the interval [a, b].

The Problem

We are given a region where 1.5≤X≤31.5 \leq X \leq 3. We need to find the area of this region using the uniform probability distribution.

Calculating the Area

To calculate the area of the region, we need to integrate the probability density function (PDF) of the uniform distribution over the interval [1.5, 3]. The PDF of the uniform distribution is given by:

f(x) = 1 / (b - a)

In this case, a = 1.5 and b = 3. Therefore, the PDF is:

f(x) = 1 / (3 - 1.5) = 1 / 1.5 = 2/3

Integrating the PDF

To find the area of the region, we need to integrate the PDF over the interval [1.5, 3]. The integral of the PDF is given by:

∫[1.5, 3] f(x) dx = ∫[1.5, 3] (2/3) dx = (2/3) ∫[1.5, 3] dx = (2/3) [x] from 1.5 to 3 = (2/3) [3 - 1.5] = (2/3) [1.5] = 2/3

Conclusion

The area of the region where 1.5≤X≤31.5 \leq X \leq 3 is given by the integral of the PDF of the uniform distribution over the interval [1.5, 3]. The integral is equal to 2/3.

Answer

The answer to the problem is:

The final answer is: 23\boxed{\frac{2}{3}}

Discussion

The uniform distribution is a continuous probability distribution where every possible outcome has an equal likelihood of occurring. In this article, we used the uniform distribution to solve a problem involving the area of a region. The area of the region was found to be 2/3.

Related Topics

  • Uniform distribution
  • Probability density function (PDF)
  • Integration
  • Area of a region

References

  • [1] "Uniform Distribution" by Wikipedia
  • [2] "Probability Density Function" by Wikipedia
  • [3] "Integration" by Wikipedia
  • [4] "Area of a Region" by Wikipedia

Further Reading

  • [1] "Uniform Distribution" by MathWorld
  • [2] "Probability Density Function" by MathWorld
  • [3] "Integration" by MathWorld
  • [4] "Area of a Region" by MathWorld

Practice Problems

  • Find the area of the region where 0≤X≤20 \leq X \leq 2 using the uniform distribution.
  • Find the area of the region where −1≤X≤1-1 \leq X \leq 1 using the uniform distribution.
  • Find the area of the region where 2≤X≤42 \leq X \leq 4 using the uniform distribution.

Conclusion

In this article, we used the uniform distribution to solve a problem involving the area of a region. The area of the region was found to be 2/3. We also discussed related topics such as the uniform distribution, probability density function (PDF), integration, and area of a region.

Introduction

In our previous article, we discussed how to use the uniform distribution to solve a problem involving the area of a region. In this article, we will answer some frequently asked questions (FAQs) about uniform distribution and area of a region.

Q: What is the uniform distribution?

A: The uniform distribution is a continuous probability distribution where every possible outcome has an equal likelihood of occurring. It is often used to model situations where all possible outcomes are equally likely.

Q: What is the probability density function (PDF) of the uniform distribution?

A: The probability density function (PDF) of the uniform distribution is given by:

f(x) = 1 / (b - a)

where a and b are the lower and upper bounds of the distribution, respectively.

Q: How do I calculate the area of a region using the uniform distribution?

A: To calculate the area of a region using the uniform distribution, you need to integrate the probability density function (PDF) of the uniform distribution over the interval [a, b]. The integral is given by:

∫[a, b] f(x) dx

Q: What is the area of the region where 1.5≤X≤31.5 \leq X \leq 3 using the uniform distribution?

A: The area of the region where 1.5≤X≤31.5 \leq X \leq 3 using the uniform distribution is given by:

∫[1.5, 3] f(x) dx = ∫[1.5, 3] (2/3) dx = (2/3) ∫[1.5, 3] dx = (2/3) [x] from 1.5 to 3 = (2/3) [3 - 1.5] = (2/3) [1.5] = 2/3

Q: How do I find the area of a region where the lower bound is less than 0?

A: If the lower bound of the region is less than 0, you need to adjust the integral accordingly. For example, if the region is −1≤X≤1-1 \leq X \leq 1, you need to integrate the PDF over the interval [-1, 1].

Q: Can I use the uniform distribution to model a situation where the outcomes are not equally likely?

A: No, the uniform distribution is only used to model situations where the outcomes are equally likely. If the outcomes are not equally likely, you need to use a different probability distribution, such as the normal distribution or the exponential distribution.

Q: What is the difference between the uniform distribution and the normal distribution?

A: The uniform distribution is a continuous probability distribution where every possible outcome has an equal likelihood of occurring. The normal distribution, on the other hand, is a continuous probability distribution that is symmetric about the mean and has a bell-shaped curve.

Q: Can I use the uniform distribution to model a situation where the outcomes are continuous?

A: Yes, the uniform distribution can be used to model a situation where the outcomes are continuous. However, you need to make sure that the lower and upper bounds of the distribution are well-defined.

Q: How do I choose the lower and upper bounds of the uniform distribution?

A: The lower and upper bounds of the uniform distribution should be chosen based on the problem you are trying to solve. For example, if you are trying to model a situation where the outcomes are between 0 and 10, you can choose the lower bound to be 0 and the upper bound to be 10.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about uniform distribution and area of a region. We hope that this article has been helpful in clarifying some of the concepts related to uniform distribution and area of a region.

Related Topics

  • Uniform distribution
  • Probability density function (PDF)
  • Integration
  • Area of a region
  • Normal distribution
  • Exponential distribution

References

  • [1] "Uniform Distribution" by Wikipedia
  • [2] "Probability Density Function" by Wikipedia
  • [3] "Integration" by Wikipedia
  • [4] "Area of a Region" by Wikipedia
  • [5] "Normal Distribution" by Wikipedia
  • [6] "Exponential Distribution" by Wikipedia

Further Reading

  • [1] "Uniform Distribution" by MathWorld
  • [2] "Probability Density Function" by MathWorld
  • [3] "Integration" by MathWorld
  • [4] "Area of a Region" by MathWorld
  • [5] "Normal Distribution" by MathWorld
  • [6] "Exponential Distribution" by MathWorld

Practice Problems

  • Find the area of the region where 0≤X≤20 \leq X \leq 2 using the uniform distribution.
  • Find the area of the region where −1≤X≤1-1 \leq X \leq 1 using the uniform distribution.
  • Find the area of the region where 2≤X≤42 \leq X \leq 4 using the uniform distribution.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about uniform distribution and area of a region. We hope that this article has been helpful in clarifying some of the concepts related to uniform distribution and area of a region.