6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables:
| \(x\) | 0 | 1 | 2 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 3 } { 5 }\) | \(\frac { 3 } { 10 }\) | \(\frac { 1 } { 10 }\) |
| \(y\) | 0 | 1 |
| \(\mathrm { P } ( Y = y )\) | \(\frac { 5 } { 8 }\) | \(\frac { 3 } { 8 }\) |
The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).
- Tabulate the probability distribution for \(Z\).
- Calculate \(\mathrm { E } ( Z )\).
- Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
- Calculate Var (3Z-4).