12.4 Exercise 35 — 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 29 and Exercise 40 from "Problems on Mathematical Expectation").

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\}

E \left[X\right] = \frac{2313}{1790} , E \left[Z\right] = \frac{1422}{895} , E \left[Z^{2}\right] = \frac{28296}{6265}

Determine $Var \left[Z\right]$ and $Cov \left[X , Z\right]$ . Check with discrete approximation.

Pfeiffer, Applied Probability, 12.4 Problems on Variance, Covariance, Linear Regression, exercise 35

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