12.4 Exercise 32 — Applied Probability

$f_{X Y} \left(t , u\right) = \frac{3}{88} \left(2 t + 3 u^{2}\right)$ for $0 \leq t \leq 2$ , $0 \leq u \leq 1 + t$ (see Exercise 25 and Exercise 37 from "Problems on Mathematical Expectation").

Z = I_{\left[0 , 1\right]} \left(X\right) 4 X + I_{\left(1 , 2\right]} \left(X\right) \left(X + Y\right)

E \left[X\right] = \frac{313}{220} , E \left[Z\right] = \frac{5649}{1760} , E \left[Z^{2}\right] = \frac{4881}{440}

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 32

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