12.4 Exercise 36 — Applied Probability

$f_{X Y} \left(t , u\right) = \frac{12}{227} \left(3 t + 2 t u\right)$ , for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{1 + t , 2\right\}$ (see Exercise 30 and Exercise 41 from "Problems on Mathematical Expectation").

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\}

E \left[X\right] = \frac{1567}{1135} , E \left[Z\right] = \frac{5774}{3405} , E \left[Z^{2}\right] = \frac{56673}{15890}

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 36

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