5. Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with A = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, A). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/A and the distribution standard deviation 1/A.
5. Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with A = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, A). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/A and the distribution standard deviation 1/A.
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5) Use R to calculate and simulate with the exponential distribution as follows.
(a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample.
(b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.

Transcribed Image Text:### Using R for Exponential Distribution Simulations
**Task:**
Calculate and simulate using the exponential distribution:
1. **Simulation with Exponential Distribution:**
- Consider an exponential random variable \( X \) with rate parameter \( \lambda = 4 \).
- Simulate 1000 independent exponential random variables using the R function `rexp(n, λ)`.
- Calculate the mean and variance of this sample.
2. **Comparative Analysis:**
- Compare the empirical results (mean and variance from the simulated data) to the theoretical distribution mean \( 1/\lambda \) and standard deviation \( 1/\lambda \).
**Instructions:**
- **R Code Example:**
To simulate the exponential random variables and compute the descriptive statistics, you can use the following R code:
```R
# Set parameters
lambda <- 4
n <- 1000
# Simulate 1000 exponential random variables
set.seed(123) # Setting seed for reproducibility
sample <- rexp(n, lambda)
# Calculate mean and variance
sample_mean <- mean(sample)
sample_variance <- var(sample)
# Theoretical mean and standard deviation
theoretical_mean <- 1 / lambda
theoretical_sd <- 1 / lambda
# Output results
cat("Sample Mean:", sample_mean, "\n")
cat("Sample Variance:", sample_variance, "\n")
cat("Theoretical Mean:", theoretical_mean, "\n")
cat("Theoretical Standard Deviation:", theoretical_sd, "\n")
```
- **Understanding Results:**
- The **mean** of an exponential distribution is \( 1/\lambda \), and the **variance** is \( 1/\lambda^2 \).
- By comparing the empirical sample mean and variance with the theoretical values, you can assess the accuracy of the simulation.
This exercise helps in understanding how simulations can approximate theoretical distributions and is a practical application of using R for statistical computations.

Transcribed Image Text:5. Use R to calculate and simulate with the exponential distribution as follows.
(a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample.
(b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.
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