6 The function \(\mathrm { F } ( t )\) is defined as follows.
$$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0
\sin ^ { 4 } t & 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi
1 & t > \frac { 1 } { 2 } \pi \end{cases}$$
- Verify that F is a (cumulative) distribution function.
The continuous random variable \(T\) has (cumulative) distribution function \(\mathrm { F } ( t )\).
- Find the lower quartile of \(T\).
- Find the (cumulative) distribution function of \(Y\), where \(Y = \sin T\), and obtain the probability density function of \(Y\).
- Find the expected value of \(\frac { 1 } { Y ^ { 3 } + 2 Y ^ { 4 } }\).