The function F(x)= Using R: a) Using R, create a user-defined function for F (named F) in the interval [0,4]. b) Using this F, find the probability that X < 2.3 | c) Using F, find the probability that X > 3.7| d) Using F, find the probability that 2.3 < X < 3.7 32 (6-x).x² for 0 ≤x ≤4 and 0 for x<0 and 1 for x > 4 is the CDF of a random variable X. e) What is the probability density function(in R code) for X between 0 and 4 ?

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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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^
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
```
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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