10.4 Exercise 14 — Applied Probability

$f_{X Y} \left(t , u\right) = \frac{12}{179} \left(3 t^{2} + u\right)$ , for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{2 , 3 - t\right\}$ (see Exercise 19 from "Problems on Random Vectors and Joint Distributions").

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

Determine $P \left(Z \leq 2\right)$ .

Pfeiffer, Applied Probability, 10.4 Problems on Functions of Random Variables, exercise 14

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