6.
The continuous random variable \(X\) has the cumulative distribution function shown below.
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < 0
\frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2
1 & x > 2
\end{array} \right.$$
The discrete random variable \(Y\) has the probability distribution shown below.
| \(y\) | 2 | 7 | 13 | 19 |
| \(\mathrm { P } ( Y = y )\) | 0.5 | 0.1 | 0.1 | 0.3 |
The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).
[0pt]
[6 marks]