6. The discrete random variable \(X\) has probability function
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l }
a ( 3 - x ) & x = 0,1,2
b & x = 3
\end{array} \right.$$
- Find \(\mathrm { P } ( X = 2 )\) and complete the table below.
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(3 a\) | \(2 a\) | | \(b\) |
- Find the value of \(a\) and the value of \(b\).
Find
- \(\mathrm { P } ( 0.5 < X < 3 )\),
- \(\mathrm { E } ( 3 X - 2 )\).
- Show that the \(\operatorname { Var } ( X ) = 1.64\)
- Calculate \(\operatorname { Var } ( 3 X - 2 )\).