find the V av iance and St andard deviation
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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Please help with 1a-1c. Thank You!
![### Understanding Continuous Random Variables
Given the function f(x) = kx, let's explore its properties as a continuous random variable over the interval \([0,2]\).
1. **Definition of f(x):**
Let \( f(x) = kx \) and \( x \) be a continuous random variable over the interval \([0,2]\).
2. **Probability Density Function (PDF):**
a) If \( f(x) \) is a probability density function of \( x \) over the interval \([0,2]\), what should be the value of \( k \)?
b) **Finding the Expected Value (Mean):**
Find the expected value (mean) of \( x \).
c) **Finding the Variance and Standard Deviation:**
Find the variance and standard deviation of \( x \).
### Detailed Solutions
#### a) Determining the Value of \( k \):
To be a PDF, \( f(x) \) must satisfy two conditions:
1. \( f(x) \geq 0 \) for all \( x \) in the interval \([0,2]\).
2. The total area under the curve \( f(x) \) over \([0,2]\) must be equal to 1, mathematically:
\[
\int_{0}^{2} f(x) dx = 1
\]
Substitute \( f(x) = kx \):
\[
\int_{0}^{2} kx \, dx = 1
\]
Solving this gives the value of \( k \).
#### b) Expected Value (Mean):
The expected value \( E[X] \) of \( x \) is calculated as:
\[
E[X] = \int_{0}^{2} x f(x) \, dx
\]
Where \( f(x) = kx \).
#### c) Variance and Standard Deviation:
The variance \( Var(X) \) is calculated as:
\[
Var(X) = E[X^2] - (E[X])^2
\]
And \( E[X^2] \) is:
\[
E[X^2] = \int_{0}^{2} x^2 f(x) \, dx
\]
Once the variance is determined, the standard deviation \( \sigma \) is:
\[
\sigma](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F059e2b6a-3014-4b4b-8bf0-49c8d5e6302d%2F231f59c5-fe52-4188-87a8-5e6b6884ec06%2Foy7pg1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Continuous Random Variables
Given the function f(x) = kx, let's explore its properties as a continuous random variable over the interval \([0,2]\).
1. **Definition of f(x):**
Let \( f(x) = kx \) and \( x \) be a continuous random variable over the interval \([0,2]\).
2. **Probability Density Function (PDF):**
a) If \( f(x) \) is a probability density function of \( x \) over the interval \([0,2]\), what should be the value of \( k \)?
b) **Finding the Expected Value (Mean):**
Find the expected value (mean) of \( x \).
c) **Finding the Variance and Standard Deviation:**
Find the variance and standard deviation of \( x \).
### Detailed Solutions
#### a) Determining the Value of \( k \):
To be a PDF, \( f(x) \) must satisfy two conditions:
1. \( f(x) \geq 0 \) for all \( x \) in the interval \([0,2]\).
2. The total area under the curve \( f(x) \) over \([0,2]\) must be equal to 1, mathematically:
\[
\int_{0}^{2} f(x) dx = 1
\]
Substitute \( f(x) = kx \):
\[
\int_{0}^{2} kx \, dx = 1
\]
Solving this gives the value of \( k \).
#### b) Expected Value (Mean):
The expected value \( E[X] \) of \( x \) is calculated as:
\[
E[X] = \int_{0}^{2} x f(x) \, dx
\]
Where \( f(x) = kx \).
#### c) Variance and Standard Deviation:
The variance \( Var(X) \) is calculated as:
\[
Var(X) = E[X^2] - (E[X])^2
\]
And \( E[X^2] \) is:
\[
E[X^2] = \int_{0}^{2} x^2 f(x) \, dx
\]
Once the variance is determined, the standard deviation \( \sigma \) is:
\[
\sigma
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