MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Please assist with all questions and add R script as well.
![**Understanding and Implementing Cumulative Distribution Function (CDF) in R**
The function \( F(x) = \frac{1}{32}(6 - x) \cdot x^2 \) for \( 0 \leq x \leq 4 \) and \( 0 \) for \( x < 0 \) and \( 1 \) for \( x > 4 \) is the cumulative distribution function (CDF) of a random variable \( X \).
**Using R:**
**a) Using R, create a user-defined function for \( F \) (named F) in the interval \([0, 4]\).**
```R
F <- function(x) {
if(x < 0) return(0)
if(x > 4) return(1)
return((1/32) * (6 - x) * x^2)
}
```
**b) Using this \( F \), find the probability that \( X < 2.3 \).**
```R
prob_X_less_than_2_3 <- F(2.3)
print(prob_X_less_than_2_3)
```
**c) Using \( F \), find the probability that \( X > 3.7 \).**
```R
prob_X_greater_than_3_7 <- 1 - F(3.7)
print(prob_X_greater_than_3_7)
```
**d) Using \( F \), find the probability that \( 2.3 < X < 3.7 \).**
```R
prob_X_between_2_3_and_3_7 <- F(3.7) - F(2.3)
print(prob_X_between_2_3_and_3_7)
```
**e) What is the probability density function (in R code) for \( X \) between 0 and 4?**
```R
f <- function(x) {
if(x < 0 || x > 4) return(0)
return((1/16) * x * (3 - x))
}
```
**f) Paste your R script in the following box**
```R
F <- function(x) {
if(x < 0) return(0)
if(x > 4) return(1)
return((1/32) * (6 - x) * x^](https://content.bartleby.com/qna-images/question/a61d77b8-3322-4869-a508-5eb685631aec/41aaf77e-2b16-4e05-b0fd-ba47009154e0/n43bh59_processed.png)
Transcribed Image Text:**Understanding and Implementing Cumulative Distribution Function (CDF) in R**
The function \( F(x) = \frac{1}{32}(6 - x) \cdot x^2 \) for \( 0 \leq x \leq 4 \) and \( 0 \) for \( x < 0 \) and \( 1 \) for \( x > 4 \) is the cumulative distribution function (CDF) of a random variable \( X \).
**Using R:**
**a) Using R, create a user-defined function for \( F \) (named F) in the interval \([0, 4]\).**
```R
F <- function(x) {
if(x < 0) return(0)
if(x > 4) return(1)
return((1/32) * (6 - x) * x^2)
}
```
**b) Using this \( F \), find the probability that \( X < 2.3 \).**
```R
prob_X_less_than_2_3 <- F(2.3)
print(prob_X_less_than_2_3)
```
**c) Using \( F \), find the probability that \( X > 3.7 \).**
```R
prob_X_greater_than_3_7 <- 1 - F(3.7)
print(prob_X_greater_than_3_7)
```
**d) Using \( F \), find the probability that \( 2.3 < X < 3.7 \).**
```R
prob_X_between_2_3_and_3_7 <- F(3.7) - F(2.3)
print(prob_X_between_2_3_and_3_7)
```
**e) What is the probability density function (in R code) for \( X \) between 0 and 4?**
```R
f <- function(x) {
if(x < 0 || x > 4) return(0)
return((1/16) * x * (3 - x))
}
```
**f) Paste your R script in the following box**
```R
F <- function(x) {
if(x < 0) return(0)
if(x > 4) return(1)
return((1/32) * (6 - x) * x^
![### Probability Density Function and Computations
The probability density function of random variable \( X \) is given by:
\[ f(x) = \frac{3}{500}(10x - x^2) \quad \text{for} \quad 0 \leq x \leq 10 \quad \text{and} \quad 0 \quad \text{otherwise.} \]
Perform the following computations using the R `integrate` function:
#### a) Find the probability that \( X > 6 \)
\[ \boxed{ }
\]
#### b) Find the probability that \( 3 < X < 7 \)
\[ \boxed{ }
\]
#### c) Find the expected value of \( X \)
\[ \boxed{ }
\]
#### d) Find the variance of \( X \)
\[ \boxed{ }
\]
#### e) Find the standard deviation of \( X \)
\[ \boxed{ }
\]
#### f) Find the probability that \( X \) is within 0.50 standard deviations of its expected value
\[ \boxed{ }
\]
#### g) In the following paste your R script for this problem
```R
# Your R script here
```](https://content.bartleby.com/qna-images/question/a61d77b8-3322-4869-a508-5eb685631aec/41aaf77e-2b16-4e05-b0fd-ba47009154e0/9s5at8s_processed.png)
Transcribed Image Text:### Probability Density Function and Computations
The probability density function of random variable \( X \) is given by:
\[ f(x) = \frac{3}{500}(10x - x^2) \quad \text{for} \quad 0 \leq x \leq 10 \quad \text{and} \quad 0 \quad \text{otherwise.} \]
Perform the following computations using the R `integrate` function:
#### a) Find the probability that \( X > 6 \)
\[ \boxed{ }
\]
#### b) Find the probability that \( 3 < X < 7 \)
\[ \boxed{ }
\]
#### c) Find the expected value of \( X \)
\[ \boxed{ }
\]
#### d) Find the variance of \( X \)
\[ \boxed{ }
\]
#### e) Find the standard deviation of \( X \)
\[ \boxed{ }
\]
#### f) Find the probability that \( X \) is within 0.50 standard deviations of its expected value
\[ \boxed{ }
\]
#### g) In the following paste your R script for this problem
```R
# Your R script here
```
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