6 The graph of the probability density function f of a random variable \(X\) is symmetrical about the line \(x = 2\). It is given that \(\mathrm { P } ( 2 < X < 5 ) = \frac { 117 } { 256 }\).
- Using only this information show that \(\mathrm { P } ( X > - 1 ) = \frac { 245 } { 256 }\).
It is now given that, for \(x\) in a suitable domain,
$$f ( x ) = k \left( 12 + 4 x - x ^ { 2 } \right) , \text { where } k \text { is a constant. }$$ - Find the value of \(k\).
- A different random variable \(X\) has probability density function \(\mathbf { g } ( x ) = \frac { 2 } { 9 } \left( 2 + x - x ^ { 2 } \right)\). The domain of \(X\) is all values of \(x\) for which \(\mathrm { g } ( x ) \geqslant 0\).
Find \(\operatorname { Var } ( X )\).
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