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 \(\mathrm { P } ( m < X < \mathrm { E } ( X ) )\).

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 \(\mathrm { P } ( X < 3 \mu - \eta )\).

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

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

A continuous random variable \(X\) has probability density function $$\mathrm{f}(x) = \begin{cases} ax^2 + bx & 1 \leq x \leq 7 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(114a + 24b = 1\) [4]

Given that \(a = \frac{1}{90}\)

1. use algebraic integration to find \(\mathrm{E}(X)\) [4]
2. find the cumulative distribution function of \(X\), specifying it for all values of \(x\) [3]
3. find \(\mathrm{P}(X > \mathrm{E}(X))\) [2]
4. use your answer to part (d) to describe the skewness of the distribution. [2]