7. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$\begin{array} { l l }
\mathrm { f } ( x ) = \frac { 2 x } { 3 } & 0 \leq x < 1
\mathrm { f } ( x ) = 1 - \frac { x } { 3 } & 1 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. }
\end{array}$$
- Sketch the graph of \(\mathrm { f } ( x )\) for all \(x\).
- Find the mean of \(X\).
- Find the standard deviation of \(X\).
- Show that the cumulative distribution function of \(X\) is given by
$$\mathrm { F } ( x ) = \frac { x ^ { 2 } } { 3 } \quad 0 \leq x < 1$$
and find \(\mathrm { F } ( x )\) for \(1 \leq x \leq 3\).