For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mean. f(x): 3 485 Ex, [3,8] a) Find the mean. μ=| (Round to three decimal places as needed.) b) Find the standard deviation. 6= (Round to three decimal places as needed.) c) Find the probability that the random variable x is within one standard deviation of the mean. The probability is (Round to three decimal places as needed.)
For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mean. f(x): 3 485 Ex, [3,8] a) Find the mean. μ=| (Round to three decimal places as needed.) b) Find the standard deviation. 6= (Round to three decimal places as needed.) c) Find the probability that the random variable x is within one standard deviation of the mean. The probability is (Round to three decimal places as needed.)
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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### Probability and Statistics: Probability Density Function
When working with probability density functions, there are key metrics you will often need to calculate:
1. **Mean (μ)**
2. **Standard Deviation (σ)**
3. **Probability that a random variable is within one standard deviation of the mean**
Consider the probability density function \( f \) defined on the random variable \( x \). Given:
\[ f(x) = \frac{3}{485} x^2, \quad [3, 8] \]
We will proceed with solving the following:
#### a) Find the mean.
\[
\mu = \boxed{\phantom{000}}
\]
(Round to three decimal places as needed.)
#### b) Find the standard deviation.
\[
\sigma = \boxed{\phantom{000}}
\]
(Round to three decimal places as needed.)
#### c) Find the probability that the random variable \( x \) is within one standard deviation of the mean.
\[
\text{The probability is } \boxed{\phantom{000}}
\]
(Round to three decimal places as needed.)
---
### Explanation of Formulas and Graphical Elements
1. **Mean (μ)**:
- The mean of a continuous random variable is found by integrating the product of the variable and its probability density function over the given interval.
\[
\mu = \int_{3}^{8} x f(x) \, dx
\]
2. **Standard Deviation (σ)**:
- The standard deviation is the square root of the variance, which is found by integrating the squared difference between the variable and the mean, all multiplied by the probability density function.
\[
\sigma^2 = \int_{3}^{8} (x - \mu)^2 f(x) \, dx
\]
3. **Probability within One Standard Deviation**:
- To find the probability that \( x \) is within one standard deviation of the mean, you calculate the integral of the probability density function from \( (\mu - \sigma) \) to \( (\mu + \sigma) \).
\[
P(\mu - \sigma \leq x \leq \mu + \sigma) = \int_{\mu - \sigma}^{\mu + \sigma} f(x) \, dx
\]
Each of these calculations involves the integration of the given probability density function within specified limits, allowing us](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bde779e-c68e-4dd4-a0b8-70b65f5ae251%2Faebeb2e2-727f-4e0e-89f5-aeee5aa472bb%2Fhwfb2vp_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Probability and Statistics: Probability Density Function
When working with probability density functions, there are key metrics you will often need to calculate:
1. **Mean (μ)**
2. **Standard Deviation (σ)**
3. **Probability that a random variable is within one standard deviation of the mean**
Consider the probability density function \( f \) defined on the random variable \( x \). Given:
\[ f(x) = \frac{3}{485} x^2, \quad [3, 8] \]
We will proceed with solving the following:
#### a) Find the mean.
\[
\mu = \boxed{\phantom{000}}
\]
(Round to three decimal places as needed.)
#### b) Find the standard deviation.
\[
\sigma = \boxed{\phantom{000}}
\]
(Round to three decimal places as needed.)
#### c) Find the probability that the random variable \( x \) is within one standard deviation of the mean.
\[
\text{The probability is } \boxed{\phantom{000}}
\]
(Round to three decimal places as needed.)
---
### Explanation of Formulas and Graphical Elements
1. **Mean (μ)**:
- The mean of a continuous random variable is found by integrating the product of the variable and its probability density function over the given interval.
\[
\mu = \int_{3}^{8} x f(x) \, dx
\]
2. **Standard Deviation (σ)**:
- The standard deviation is the square root of the variance, which is found by integrating the squared difference between the variable and the mean, all multiplied by the probability density function.
\[
\sigma^2 = \int_{3}^{8} (x - \mu)^2 f(x) \, dx
\]
3. **Probability within One Standard Deviation**:
- To find the probability that \( x \) is within one standard deviation of the mean, you calculate the integral of the probability density function from \( (\mu - \sigma) \) to \( (\mu + \sigma) \).
\[
P(\mu - \sigma \leq x \leq \mu + \sigma) = \int_{\mu - \sigma}^{\mu + \sigma} f(x) \, dx
\]
Each of these calculations involves the integration of the given probability density function within specified limits, allowing us
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