**Q1: Simulate 10,000 samples from an Exp(3) distribution, each of size n = 10, using the inverse transform method. You should save the generated values in R for later use. You must use the command set.seed(3202) to initialise the random number generator, before you start the simulation.**

**Answer:**

The task is to simulate 10,000 samples from an Exponential distribution with a rate parameter of 3 (Exp(3)), and each sample should consist of 10 values. The code also requires setting the seed to ensure reproducibility.

Here’s the step-by-step explanation of the code and solution:

**Setting the Seed**:`set.seed(3202)`

is used to set the seed for the random number generator. Setting a seed ensures that the random numbers generated are reproducible. In other words, if you run the code with the same seed, you will get the same set of random numbers each time.**Simulation Parameters**:

- Number of Samples (
`num_samples`

): 10,000 samples are to be generated. - Rate Parameter (
`rate_parameter`

): The rate parameter of the Exponential distribution is set to 3. - Sample Size (
`sample_size`

): Each sample is to consist of 10 values.

**Initialize Data Structure**:`samples <- numeric(num_samples * sample_size)`

initializes an empty numeric vector named`samples`

to store the simulated values. The length of this vector will be 10,000 (number of samples) multiplied by 10 (sample size), resulting in 100,000 elements.**Generating Samples using Inverse Transform Method**:

- The code uses a for loop to generate each sample.
- Inside the loop,
`runif(sample_size)`

generates`sample_size`

(which is 10) random uniform numbers between 0 and 1. These uniform random numbers will be used in the inverse transform method to generate Exponential random variables. `x <- (-1 / rate_parameter) * log(1 - u)`

is the inverse transform method formula for generating Exponential random variables. It takes the random uniform numbers`u`

, applies the formula, and stores the result in`x`

. This generates one sample of size 10.- The
`samples`

vector is updated to store the generated values in sequence.

**Saving the Generated Samples**:`save(samples, file = "exp_samples.RData")`

saves the generated samples in a file named “exp_samples.RData” for later use. This is useful if you want to work with the simulated data in future analyses without having to re-run the simulation.

In summary, the code initializes a random seed for reproducibility, simulates 10,000 samples of size 10 from an Exponential distribution with a rate parameter of 3 using the inverse transform method, and saves the generated samples for later use. This is a common procedure in statistical simulations when you need to generate data for analysis or modeling.

```
# Set the seed for reproducibility
set.seed(3202)
# Number of samples
num_samples <- 10000
# Exponential rate parameter
rate_parameter <- 3
# Sample size
sample_size <- 10
# Initialize an empty vector to store the samples
samples <- numeric(num_samples * sample_size)
# Generate samples using the inverse transform method
for (i in 1:num_samples) {
u <- runif(sample_size) # Generate uniform random numbers
x <- (-1 / rate_parameter) * log(1 - u) # Inverse transform method
samples[((i - 1) * sample_size + 1):(i * sample_size)] <- x
}
# Save the generated samples for later use
save(samples, file = "exp_samples.RData")
```

Above code sets the seed, specifies the parameters of the Exponential distribution, and then uses a for loop to generate the samples using the inverse transform method. Finally, it saves the generated samples in a file named “exp_samples.RData” for later use.

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