7 A continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } ( 4 - x ) & 1 \leqslant x \leqslant 3
\frac { 1 } { 6 } & 3 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$

1. Draw the graph of f on the grid on page 6 .
2. Prove that the mean of \(X\) is \(2 \frac { 5 } { 9 }\).
3. Calculate the exact value of:
   1. \(\mathrm { P } ( X > 2.5 )\);
   2. \(\mathrm { P } ( 1.5 < X < 4.5 )\);
   3. \(\mathrm { P } ( X > 2.5\) and \(1.5 < X < 4.5 )\);
   4. \(\mathrm { P } ( X > 2.5 \mid 1.5 < X < 4.5 )\).
      \includegraphics[max width=\textwidth, alt={}, center]{bc21c177-6cd8-4c79-8782-d17f0238ce17-6_1340_1363_317_383}