6. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x < 1
x - \frac { 1 } { 2 } & 1 \leqslant x \leqslant k
0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that \(k = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find \(\mathrm { P } ( 0.5 < X < 1.5 )\).
5. Write down the median of \(X\) and the mode of \(X\).
6. Describe the skewness of the distribution of \(X\). Give a reason for your answer.