Question

In today's lecture, several of you were suspicious of my claim that the E (i.e., distribution mean) of the ith element of a sample of size n is equal to the population mean, i.e. E[x_i] = E[x]. To convince you, write code to - take 10^7 samples of size 50 from a normal distribution with mu = 2 and sigma = 3, - select the 3rd element in each of the 10^7 samples, and store them in an array called x3. - compute the mean of x3. Convinced?! Note: your code will take a while to run because we are taking 10^7 samples (i.e., a large number). But that's what it takes to compute the E of something. E is a distribution mean, and the only way to approximate it numerically is to take a very large sample.

Answer 1

I hope Rstudio is fine for the exercise, since you didn't mention a specific language.

x3=c() #Initialize the vector which will contain the 3rd element of each sample.
for (i in 1:10000000) {
sample_vector=rnorm(50,2,3) #This gives us a random sample of 50 elements from a normal distribution with mean=2 and standard deviation=3
x3=c(x3,sample_vector[3]) #This simply appends the 3rd element of the sample generated to the x3 vector

}
mean(x3) #This calculates the mean.

Note: The output will be different each time you run the code. Sample output I got:

Remember that output will differ every time you run this code. You may use set.seed to fix a simulation