7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { x } { 15 } , & 0 \leq x \leq 2
\frac { 2 } { 15 } , & 2 < x < 7
\frac { 4 } { 9 } - \frac { 2 x } { 45 } , & 7 \leq x \leq 10
0 , & \text { otherwise } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. 1. Find expressions for the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
   2. Show that for \(2 < x < 7 , \mathrm {~F} ( x ) = \frac { 2 x } { 15 } - \frac { 2 } { 15 }\).
   3. Specify \(\mathrm { F } ( x )\) for \(x < 0\) and for \(x > 10\).
3. Find \(\mathrm { P } ( X \leq 8.2 )\).
4. Find, to 3 significant figures, \(\mathrm { E } ( X )\).