12.4 Exercise 34 — Applied Probability

$f_{X Y} \left(t , u\right) = \frac{3}{23} \left(t + 2 u\right)$ for $0 \leq t \leq 2$ , $0 \leq u \leq \max \left\{2 - t , t\right\}$ (see Exercise 28 and Exercise 39 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 , M = \left\{\left(t , u\right) : \max \left(t , u\right) \leq 1\right\}

E \left[X\right] = \frac{53}{46} , E \left[Z\right] = \frac{175}{92} , E \left[Z^{2}\right] = \frac{2063}{460}

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

Pfeiffer, Applied Probability, 12.4 Problems on Variance, Covariance, Linear Regression, exercise 34

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