3 The discrete random variable \(X\) has the following probability distribution
| \(\boldsymbol { x }\) | 1 | 2 | 4 | 9 |
| \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\) | 0.2 | 0.4 | 0.35 | 0.05 |
The continuous random variable \(Y\) has the following probability density function
$$\mathrm { f } ( y ) = \begin{cases} \frac { 1 } { 64 } y ^ { 3 } & 0 \leq y \leq 4
0 & \text { otherwise } \end{cases}$$
Given that \(X\) and \(Y\) are independent, show that \(\mathrm { E } \left( X ^ { 2 } + Y ^ { 2 } \right) = \frac { 1327 } { 60 }\)