7. The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} k x ( 5 - x ) , & 0 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
- Show that \(k = \frac { 3 } { 56 }\).
- Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
- Evaluate \(\mathrm { E } ( X )\).
- Find the modal value of \(X\).
- Verify that the median value of \(X\) lies between 2.3 and 2.5.
- Comment on the skewness of \(X\). Justify your answer.
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