Interquartile range calculation

5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2 ,
\frac { ( x - 2 ) ^ { 2 } } { 12 } & 2 \leqslant x < 4 ,
1 - \frac { ( 8 - x ) ^ { 2 } } { 24 } & 4 \leqslant x \leqslant 8 ,
1 & x > 8 . \end{cases}$$ (c) Find the exact value of the interquartile range of \(X\).

6 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 60 } \left( 16 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6
1 & x > 6 \end{cases}$$

1. Find the interquartile range of \(X\).
2. Find \(\mathrm { E } \left( X ^ { 3 } \right)\).
   The random variable \(Y\) is such that \(Y = \sqrt { X }\).
3. Find the probability density function of \(Y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X > 2 )\).
2. Find the interquartile range of \(X\).