Consider the population described by the probability dis-tribution shown below.
x p1x2
1 2 3 4 5 .3
.2 .2 .2 .1
The random variable x is observed twice. If these observa-tions are independent, verify that the different samples of size 2 and their probabilities are as shown below.
Sample Probability 1, 1
1, 2 1, 3 1, 4 1, 5 2, 1 2, 2 2, 3 2, 4 2, 5 3, 1 3, 2 3, 3
.04 .06 .04 .04 .02 .06 .09 .06 .06 .03 .04 .06 .04
Sample Probability 3, 4
3, 5 4, 1 4, 2 4, 3 4, 4 4, 5 5, 1 5, 2 5, 3 5, 4 5, 5
.04 .02 .04 .06 .04 .04 .02 .02 .03 .02 .02 .01
5.4 Refer to Exercise 5.3 and find E1x2 = m. Then use the sampling distribution of x found in Exercise 5.3 to find the expected value of x. Note that E1x2 = m.
5.5 Refer to Exercise 5.3. Assume that a random sample of n = 2 measurements is randomly selected from the population.
a. List the different values that the sample median m may assume and find the probability of each. Then give the sampling distribution of the sample median.
b. Construct a probability histogram for the sampling distribution of the sample median and compare it with the probability histogram for the sample mean (Exercise 5.3, part b).
