6. 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.
a Show that \(k = 0.25\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
b Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\)
c Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
d Find \(\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)\)