10 The probability density function of the continuous random variable \(X\) is given by
\(f ( x ) = \begin{cases} k x ^ { m } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise, } \end{cases}\)
where \(a , k\) and \(m\) are positive constants.
- Show that \(k = \frac { m + 1 } { a ^ { m + 1 } }\).
- Find the cumulative distribution function of \(X\) in terms of \(x , a\) and \(m\).
- Given that \(\mathrm { P } \left( \frac { 1 } { 4 } a < X < \frac { 1 } { 2 } a \right) = \frac { 1 } { 10 }\),
- show that \(2 p ^ { 2 } - 10 p + 5 = 0\), where \(p = 2 ^ { m }\),
- find the value of \(m\).
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