(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 = 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 = 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")
Using R, Exercise 4 (CLT Simulation) For this exercise we will simulate from the exponential distribution....
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1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...