- The continuous random variable \(X\) has probability density function
$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0
\frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
where \(c\) is a positive constant.
- Show that \(c = \frac { 1 } { 12 }\)
- Sketch the probability density function.
- Explain why the mode of \(X = 0\)
- Find the cumulative distribution function of \(X\), for all values of \(x\)
- Find, to 3 significant figures, the value of \(d\) such that \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)
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