Interquartile range calculation

6 The continuous random variable \(Y\) has probability density function given by $$f ( y ) = \begin{cases} - \frac { 1 } { 4 } y & - 2 \leqslant y < 0
\frac { 1 } { 4 } y & 0 \leqslant y \leqslant 2
0 & \text { otherwise. } \end{cases}$$ Find

1. the interquartile range of \(Y\),
2. \(\operatorname { Var } ( Y )\),
3. \(\mathrm { E } ( | Y | )\).

3. A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 2
k \left( 1 - \frac { x } { 6 } \right) & 2 < x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 4 }\)
2. Write down the mode of \(X\).
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find the upper quartile of \(X\).

6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6
0 , & \text { otherwise } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Find \(\mathrm { E } ( X )\).
3. Show that \(\operatorname { Var } ( X ) = \frac { 11 } { 9 }\).
4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
5. Show that the interquartile range of \(X\) is \(2 ( \sqrt { } 7 - \sqrt { 3 } )\). END