14.2 Exercise 16 — Applied Probability

$f_{X Y} \left(t , u\right) = I_{\left[0 , 1\right]} \left(t\right) \frac{3}{8} \left(t^{2} + 2 u\right) + I_{\left(1 , 2\right]} \left(t\right) \frac{9}{14} t^{2} u^{2}$ , for $0 \leq u \leq 1$ . (see Exercise 32 from "Problems on Mathematical Expectation", and the exercise).

f_{X} \left(t\right) = I_{\left[0 , 1\right]} \left(t\right) \frac{3}{8} \left(t^{2} + 1\right) + I_{\left(1 , 2\right]} \left(t\right) \frac{3}{14} t^{2}

Z = I_{M} \left(X , Y\right) X + I_{M^{c}} \left(X , Y\right) X Y , M = \left\{\left(t , u\right) : u \leq \min \left(1 , 2 - t\right)\right\}

Pfeiffer, Applied Probability, 14.2 Problems on Conditional Expectation, Regression, exercise 16

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