11.3 Exercise 31 — Applied Probability

(See Exercise 21 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{2}{13} \left(t + 2 u\right)$ , for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{2 t , 3 - t\right\}$ .

f_{X} \left(t\right) = I_{\left[0 , 1\right]} \left(t\right) \frac{12}{13} t^{2} + I_{\left(1 , 2\right]} \left(t\right) \frac{6}{13} \left(3 - t\right)

f_{Y} \left(u\right) = I_{\left[0 , 1\right]} \left(u\right) \left(\frac{4}{13} + \frac{8}{13} u - \frac{9}{52} u^{2}\right) + I_{\left(1 , 2\right]} \left(u\right) \left(\frac{9}{13} + \frac{6}{13} u - \frac{21}{52} u^{2}\right)

Pfeiffer, Applied Probability, 11.3 Problems on Mathematical Expectation, exercise 31

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