6 The continuous random variable $X$ has probability density function f given by

$$f ( x ) = \begin{cases} \frac { 3 } { 28 } \left( e ^ { \frac { 1 } { 2 } x } + 4 e ^ { - \frac { 1 } { 2 } x } \right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise } \end{cases}$$

(a) Find the cumulative distribution function of $X$.\\

The random variable $Y$ is defined by $Y = e ^ { \frac { 1 } { 2 } ( X ) }$.\\
(b) Find the probability density function of $Y$.\\

(c) Find the 30th percentile of $Y$.\\

(d) Find $\mathrm { E } \left( Y ^ { 4 } \right)$.\\

If you use the following page to complete the answer to any question, the question number must be clearly shown.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q6}}