7 The continuous random variable \(T\) has probability density function given by
$$f ( t ) = \begin{cases} 4 t ^ { 3 } & 0 < t \leqslant 1
0 & \text { otherwise } \end{cases}$$
- Obtain the cumulative distribution function of \(T\).
- Find the cumulative distribution function of \(H\), where \(H = \frac { 1 } { T ^ { 4 } }\), and hence show that the probability density function of \(H\) is given by \(\mathrm { g } ( h ) = \frac { 1 } { h ^ { 2 } }\) over an interval to be stated.
- Find \(\mathrm { E } \left( 1 + 2 H ^ { - 1 } \right)\).