Derived random variables from samples

7 The discrete random variable \(X\) can take the values 0,1 and 2 with equal probabilities.
The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\), and the random variables \(Y\) and \(Z\) are defined as follows:
\(Y\) is the smaller of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal; \(Z = \left| X _ { 1 } - X _ { 2 } \right|\).

1. Draw up a table giving the joint distribution of \(Y\) and \(Z\).
2. Find \(P ( Y = 0 \mid Z = 0 )\).
3. Find \(\operatorname { Cov } ( Y , Z )\).

5 The independent discrete random variables \(U\) and \(V\) can each take the values 1, 2 and 3, all with probability \(\frac { 1 } { 3 }\). The random variables \(X\) and \(Y\) are defined as follows: $$X = | U - V | , Y = U + V .$$

1. In the Printed Answer Book complete the table showing the joint probability distribution of \(X\) and \(Y\).
2. Find \(\operatorname { Cov } ( X , Y )\).
3. State with a reason whether \(X\) and \(Y\) are independent.
4. Find \(\mathrm { P } ( Y = 3 \mid X = 1 )\).