4. A discrete random variable \(X\) has probability function
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l }
k ( 2 - x ) & x = 0,1
k ( 3 - x ) & x = 2,3
k ( x + 1 ) & x = 4
0 & \text { otherwise }
\end{array} \right.$$
where \(k\) is a constant.
- Show that \(k = \frac { 1 } { 9 }\)
Find the exact value of
- \(\mathrm { P } ( 1 \leqslant X < 4 )\)
- \(\mathrm { E } ( X )\)
- \(\mathrm { E } \left( X ^ { 2 } \right)\)
- \(\operatorname { Var } ( 3 X + 1 )\)