11.3 Exercise 30 — Applied Probability

(See Exercise 20 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{12}{227} \left(3 t + 2 t u\right)$ , for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{1 + t , 2\right\}$ .

f_{X} \left(t\right) = I_{\left[0 , 1\right]} \left(t\right) \frac{12}{227} \left(t^{3} + 5 t^{2} + 4 t\right) + I_{\left(1 , 2\right]} \left(t\right) \frac{120}{227} t

f_{Y} \left(u\right) = I_{\left[0 , 1\right]} \left(u\right) \frac{24}{227} \left(2 u + 3\right) + I_{\left(1 , 2\right]} \left(u\right) \frac{6}{227} \left(2 u + 3\right) \left(3 + 2 u - u^{2}\right)

= I_{\left[0 , 1\right]} \left(u\right) \frac{24}{227} \left(2 u + 3\right) + I_{\left(1 , 2\right]} \left(u\right) \frac{6}{227} \left(9 + 12 u + u^{2} - 2 u^{3}\right)

Pfeiffer, Applied Probability, 11.3 Problems on Mathematical Expectation, exercise 30

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