- The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by
$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3
k ( 4 - x ) & 3 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
- Show that \(k = \frac { 11 } { 12 }\)
- Sketch \(\mathrm { f } ( x )\) for \(1 \leqslant x \leqslant 4\)
- Write down the mode of \(X\)
Given that \(\mathrm { E } ( X ) = \frac { 25 } { 9 }\)
- use algebraic integration to find \(\operatorname { Var } ( X )\), giving your answer to 3 significant figures.
The cumulative distribution function of \(X\) is given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r }
0 & x < 1
\frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3
\frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4
1 & x > 4
\end{array} \right.$$ - Find the exact value of \(C\)
- Find the exact value of \(d\)
- Calculate, to 3 significant figures, the upper quartile of \(X\)
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