12.4 Exercise 37 — Applied Probability

(See the exercise, and Exercises 9 and 10 from "Problems on Functions of Random Variables"). For the pair $\left\{X , Y\right\}$ in the exercise, let

Z = g \left(X , Y\right) = 3 X^{2} + 2 X Y - Y^{2}

W = h \left(X , Y\right) = \left\{\begin{matrix} X & \text{for} X + Y \leq 4 \\ 2 Y & \text{for} X + Y > 4 \end{matrix}\right = I_{M} \left(X , Y\right) X + I_{M^{c}} \left(X , Y\right) 2 Y

Determine the joint distribution for the pair $\left\{Z , W\right\}$ and determine the regression line of W on Z.

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