Suppose that a random variableXhas the following probability distribution:

x=0 Pr(X=x)=0.2

x=1 Pr(X=x)=0.1

x=4 Pr(X=x)=0.7

(a) What is the expected value ofX,E(X), and the variance ofX,V ar(X)?

(b) Suppose you took samples of size 2 from this distribution (assume that thedraws are independent) and calculated the sample mean for each sample.Derive the probability distribution for the sample mean.

(c) What is the expected value of the distribution found in part (b)?

(d) Using precisely the same reasoning as for sample means, we can draw in-dependent samples of size 2 and calculate the sample variance,s2for eachsample. Derive the probability distribution for the sample variances2.

(e) What is the expected value of the distribution found in part (d)? Why is thiswhat you would have expected?