The probability density function of the random variable \(X\) is given by
$$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x \leqslant 2 0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
Show that \(k = \frac { 3 } { 16 }\).
Find \(\mathrm { E } ( X )\).
The random variable \(Y\) has the following properties.
\(Y\) takes values between 0 and 5 only.
The probability density function of \(Y\) is symmetrical.
Given that \(\mathrm { P } ( Y < a ) = 0.2\), find \(\mathrm { P } ( 2.5 < Y < 5 - a )\) illustrating your method with a sketch on the axes provided.
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If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.