12.4 Exercise 33 — Applied Probability

$f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{1 , 2 - t\right\}$ (see Exercise 27 and Exercise 38 from "Problems on Mathematical Expectation").

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\}

E \left[X\right] = \frac{52}{55} , E \left[Z\right] = \frac{16}{55} , E \left[Z^{2}\right] = \frac{39}{308}

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 33

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