1. In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective,….

## Find the joint probability mass function of X and Y.

1. Refer to Exercise 31. Assume the first card is not replaced before the second card is drawn.

a. Find the joint probability mass function of X and Y.

b. Find the marginal probability mass functions pX (x) and pY (y).

c. Find µX and µY .

d. Find µXY.

e. Find Cov(X,Y)

2. Blood is taken from each of n individuals to be tested for a certain disease. Rather than test each sample separately, a pooled method is used in an attempt to reduce the number of tests needed. Part of each blood sample is taken, and these parts are combined to form a pooled sample. The pooled sample is then tested. If the result is negative, then none of the n individuals has the disease, and no further tests are needed. If the pooled sample tests positive, then each individual is tested to see which of them have the disease.

a. Let X represent the number of tests that are carried out.What are the possible values of X?

b. Assume that n = 4 individuals are to be tested, and the probability that each has the disease, independent of the others, is p = 0.1.

c. Find µX . Repeat part (b) with n = 6and p = 0.2.

d. Express µX as a function of n and p.

e. The pooled method is more economical than performing individual tests if µX < n.=”” suppose=”” n=”10.” for=”” what=”” values=”” of=”” p=”” is=”” the=”” pooled=”” method=”” more=”” economical=”” than=”” performing=”” n=”” individual=””>