7.39. The following example illustrates that p = 0 does not imply independence. Suppose that (X, Y) has a joint probability distribution given by Table 7.1.

(a) Show that E(XY) = E(X)E(Y) and hence p = 0.

(b) Indicate why X and Y are not independent.

(c) Show that this example may be generalized as follows. The choice of the number is not crucial. What is important is that all the circled values are the same, all the boxed values are the same, and the center value equals zero.

Y -1 X 0 1 TABLE 7.1