Interquartile range calculation

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

The continuous random variable \(X\) has probability density function f(\(x\)) given by $$\text{f}(x) = \begin{cases} \frac{1}{20}x^3, & 1 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$$

1. Sketch f(\(x\)) for all values of \(x\). [3]
2. Calculate E(\(X\)). [3]
3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places. [3]
4. Show that for \(1 \leq x \leq 3\), P(\(X \leq x\)) is given by \(\frac{1}{80}(x^4 - 1)\) and specify fully the cumulative distribution function of \(X\). [5]
5. Find the interquartile range for the random variable \(X\). [4]

Some statisticians use the following formula to estimate the interquartile range: $$\text{interquartile range} = \frac{4}{3} \times \text{standard deviation}.$$

1. Use this formula to estimate the interquartile range in this case, and comment. [2]

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

1. the interquartile range of \(Y\), [4]
2. Var\((Y)\), [5]
3. E\((|Y|)\). [4]