Expected value: Mean: Variance: Standard Deviation: > Next Question
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Exploring Probability Density Functions
**Problem:**
Given the probability density function \( f(z) = \frac{1}{3} \) over the interval \([3, 6]\), find the expected value, the mean, the variance, and the standard deviation.
**Solution:**
To solve this problem, follow these steps:
1. **Identify the Probability Density Function (PDF):**
The PDF is given as \( f(z) = \frac{1}{3} \).
2. **Define the Interval:**
The interval is \([3, 6]\).
3. **Calculate the Expected Value (also known as the Mean):**
The expected value \( E(Z) \) for a continuous uniform distribution is given by:
\[
E(Z) = \frac{a + b}{2}
\]
where \( a \) and \( b \) are the interval endpoints.
4. **Calculate the Variance:**
The variance \( \text{Var}(Z) \) for a continuous uniform distribution is given by:
\[
\text{Var}(Z) = \frac{(b - a)^2}{12}
\]
5. **Calculate the Standard Deviation:**
The standard deviation is the square root of the variance:
\[
\text{SD}(Z) = \sqrt{\text{Var}(Z)}
\]
**Interactive Elements:**
- **Expected Value:**
- [Input Box]
- **Mean:**
- [Input Box]
- **Variance:**
- [Input Box]
- **Standard Deviation:**
- [Input Box]
[Next Question Button]
This problem provides an opportunity to practice calculations related to continuous uniform distributions, an important concept in probability and statistics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F634845f0-1267-4369-8c1c-17e520d5439c%2Ff362a954-1e0b-4daf-b7e0-2446e7d06516%2Fjg415p7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exploring Probability Density Functions
**Problem:**
Given the probability density function \( f(z) = \frac{1}{3} \) over the interval \([3, 6]\), find the expected value, the mean, the variance, and the standard deviation.
**Solution:**
To solve this problem, follow these steps:
1. **Identify the Probability Density Function (PDF):**
The PDF is given as \( f(z) = \frac{1}{3} \).
2. **Define the Interval:**
The interval is \([3, 6]\).
3. **Calculate the Expected Value (also known as the Mean):**
The expected value \( E(Z) \) for a continuous uniform distribution is given by:
\[
E(Z) = \frac{a + b}{2}
\]
where \( a \) and \( b \) are the interval endpoints.
4. **Calculate the Variance:**
The variance \( \text{Var}(Z) \) for a continuous uniform distribution is given by:
\[
\text{Var}(Z) = \frac{(b - a)^2}{12}
\]
5. **Calculate the Standard Deviation:**
The standard deviation is the square root of the variance:
\[
\text{SD}(Z) = \sqrt{\text{Var}(Z)}
\]
**Interactive Elements:**
- **Expected Value:**
- [Input Box]
- **Mean:**
- [Input Box]
- **Variance:**
- [Input Box]
- **Standard Deviation:**
- [Input Box]
[Next Question Button]
This problem provides an opportunity to practice calculations related to continuous uniform distributions, an important concept in probability and statistics.
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