7 The continuous random variable \(X\) has (cumulative) distribution function given by
$$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1
1 - \frac { 1 } { x ^ { 4 } } & x \geqslant 1 \end{cases}$$
- Find the (cumulative) distribution function, \(\mathrm { G } ( y )\), of the random variable \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
- Hence show that the probability density function of \(Y\) is given by
$$g ( y ) = \begin{cases} 2 y & 0 < y \leqslant 1
0 & \text { otherwise } \end{cases}$$ - Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).