In Sec. 11.2.2, and specifically in Fig. 11.1, we considered the question of whether CRN w...

In Sec. 11.2.2, and specifically in Fig. 11.1, we considered the question of whether CRN would induce the desired positive correlation for a given pair of alternative configurations, or whether it might backfire. Consider the following simple Monte Carlo examples, where U represents a random number:

(1) X_1j = U² and X_2j = U³

---

(2) X_1j = U² and X_2j = (1 – U)³

(a) Sketch the graphs of the responses in both examples.

---

(b) For each example, analytically find Cov(X_1j, X_2j).

---

(c) For each example, analytically calculate Var(X_1j – X_2j) under both independent sampling and CRN.

---

(d) Confirm your calculations in (b) and (c) empirically by designing and carrying out a small simulation study.

FIGURE 11.1 Model responses for CRN to work (top row) and backfire (bottom row).