Consider an infinite population with 25% of the elements having the value 1, 25% the value 2, 25% the value 3, 25% the value 4. If X is the value of a randomly selected item, then X is a discrete random variable whose possible values are 1, 2, 3, 4.

a) Find the population mean and population variance ² for the random variable X.

b) List all 16 possible distinguishable samples of size 2, and for each calculate the value of the sample mean. Represent the value of the sample mean (X bar) using a probability histogram (use one bar for each of the possible values for (X bar)). Note that although this is a very small sample, the distribution of (X bar) does not look like the population distribution and has the general shape of the normal distribution.

c) Calculate the mean and variance of the distribution of (X bar) and show that, as expected, they are equal to and²/n, respectively.