5. The discrete random variable \(X\) has probability function
$$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2
k ( x - 2 ) , & x = 3
0 , & \text { otherwise } \end{cases}$$
where \(k\) is a positive constant.
- Show that \(k = 0.25\).
- Find \(\mathrm { E } ( X )\) and show that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.5\).
- Find \(\operatorname { Var } ( 3 X - 2 )\).
Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
- Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\).
- Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
- Find \(\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)\).