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10.4 Exercise 12

Avg difficulty 2.5

$f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{1 , 2 - t\right\}$ (see Exercise 17 from "Problems on Random Vectors and Joint Distributions").

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

Determine $P \left(Z \leq 1 / 4\right)$ .

Pfeiffer, Applied Probability, 10.4 Problems on Functions of Random Variables, exercise 12

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