Compare mean and median using probability

2 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { 3 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
3. Hence explain whether the mean of \(X\) is less than, equal to or greater than the median of \(X\).

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

1. Show that \(k = \frac { 1 } { \ln 3 }\).
2. Show that \(\mathrm { E } ( X ) = 3.64\), correct to 3 significant figures.
3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < \mathrm { E } ( X ) )\).

6 The average speed of a bus, \(x \mathrm {~km} \mathrm {~h} ^ { - 1 }\), on a certain journey is a continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 20 \leqslant x \leqslant 28
0 & \text { otherwise } \end{cases}$$

1. Show that \(k = 70\).
2. Find \(\mathrm { E } ( X )\).
3. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
4. Hence determine whether the mean is greater or less than the median.

4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } 3 k & 0 \leqslant x < 1
k x ( 4 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Sketch f (x).
2. Write down the mode of \(X\). Given that \(\mathrm { E } ( X ) = \frac { 29 } { 16 }\)
3. describe, giving a reason, the skewness of the distribution.
4. Use integration to find the value of \(k\).
5. Write down the lower quartile of \(X\). Given also that \(\mathrm { P } ( 2 < X < 3 ) = \frac { 11 } { 36 }\)
6. find the exact value of \(\mathrm { P } ( X > 3 )\).

1. The continuous random variable \(Y\) has probability density function

$$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise } \end{array} \right.$$

1. Show that the mode of \(Y\) is 1 , justifying your reasoning. Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
2. determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer. Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
3. find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1
\frac { 3 - x } { 4 } & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Explain why the value of \(\eta\), the median of \(X\), is 1 .
3. Show that the value of \(\mu\), the mean of \(X\), is \(\frac { 13 } { 12 }\).
4. Find \(\mathrm { P } ( X < 3 \mu - \eta )\).