Question

Using R,

Exercise 4 (CLT Simulation) For this exercise we will simulate from the exponential distribution. If a random variable X has an exponential distribution with rate parameter A, the pdf of X can be written for z 2 0 Also recall, (a) This exercise relies heavily on generating random observations. To make this reproducible we will set a seed for the randomization. Alter the following code to make birthday store your birthday in the format yyyymmdd. For example, William Gosset, better known as Student, was born on June 13, 1876, so he would ise: birthday18760613 set.seed(birthday) (b) Simulate 10000 samples of size 5 from an exponential distribution with λ-2. Store the mean of each sample in a vector. Plot a histogram of these sample means. (Be sure to give it a title, and label the axes appropriately.) Based on the histogram, do you think the central limit theorem applies here? (c) Simulate 10000 samples of size 100 from an exponential distribution with λ-2. Store the mean of each sample in a vector. Plot a histogram of these sample means. (Be sure to give it a title, and label the axes appropriately.) Based on the histogram, do you think the central limit theorem applies here? (d) We just repeated ourselves, so that means we probably should be writing a function. Write a function called sim xbars_exp which takes three inputs: The number of samples to simulate. .The sample size The rate parameter of an exponential distribution. The function should output a vector of sample means which are the result of sampling from an exponential distribution as specified by the inputs. Use your function to simulate 25000 samples of size 50 from an exponential distribution with λ-3. Store the mean of each sample in a vector. Plot a histogram of these sample means. (Be sure to give it a title, and label the axes appropriately.)

Answer 1

(a)

We will use birthdate as 20010610 as shown in below code.

birthday = 20010610
set.seed(birthday)

(b)

Below code simulate 10,000 samples of size 5 from an exponential distribution with $\lambda$ = 2. The shape of the histogram is not symmetrical, as it is skewed to the right. Based on histogram, the central limit theorem does not applies.

birthday = 20010610
set.seed(birthday)
x = numeric()
for (i in 1:10000) {
samples = rexp(5, rate = 2)
x = c(x, mean(samples))
}
hist(x, xlab = "Sample mean of size 5", main = "Histogram of sample means of size 5")

(c)

Below code simulate 10,000 samples of size 100 from an exponential distribution with $\lambda$ = 2. The histogram shape is symmetrical. Thus, based on histogram, the central limit theorem applies.

birthday = 20010610
set.seed(birthday)
x = numeric()
for (i in 1:10000) {
samples = rexp(100, rate = 2)
x = c(x, mean(samples))
}
hist(x, xlab = "Sample mean of size 100", main = "Histogram of sample means of size 100")

(d)

Below is the function sim_xbars_exp which will take the simulation size, sample size and rate parameter to generate sample means of the exponential distribution.

sim_xbars_exp = function(simulation_size, sample_size, lambda) {
birthday = 20010610
set.seed(birthday)
x = numeric()
for (i in 1:simulation_size) {
samples = rexp(sample_size, rate = lambda)
x = c(x, mean(samples))
}
return(x)
}

Calling the function and storing the mean vector in y and then plotting the histogram.

y = sim_xbars_exp(25000, 50, 3)
hist(y, xlab = "Sample mean of size 50", main = "Histogram of sample means of size 50")

Histogram of sample means of size 50 0.2 0.3 0.4 0.5 0.6 Sample mean of size 50