Find parameter from median

6 A function f is defined by $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { a ^ { 3 } } & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that f is a probability density function for all positive values of \(a\).
   The random variable \(X\) has probability density function f and the median of \(X\) is 2 .
2. Show that \(a = 2.52\), correct to 3 significant figures.
3. Find \(\mathrm { E } ( X )\).

10 The probability density function of the continuous random variable \(X\) is given by
\(f ( x ) = \begin{cases} k x ^ { m } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise, } \end{cases}\)
where \(a , k\) and \(m\) are positive constants.

1. Show that \(k = \frac { m + 1 } { a ^ { m + 1 } }\).
2. Find the cumulative distribution function of \(X\) in terms of \(x , a\) and \(m\).
3. Given that \(\mathrm { P } \left( \frac { 1 } { 4 } a < X < \frac { 1 } { 2 } a \right) = \frac { 1 } { 10 }\),
   1. show that \(2 p ^ { 2 } - 10 p + 5 = 0\), where \(p = 2 ^ { m }\),
   2. find the value of \(m\). \section*{END OF QUESTION PAPER}

10 The continuous random variable \(X\) has probability density function given by
\(f ( x ) = \begin{cases} \frac { 4 } { 15 } \left( \frac { a } { x ^ { 2 } } + 3 x ^ { 2 } - \frac { 7 } { 2 } \right) & 1 \leqslant x \leqslant 2 ,
0 & \text { otherwise, } \end{cases}\)
where \(a\) is a positive constant.

1. Find the cumulative distribution function of \(X\) in terms of \(a\).
2. Hence or otherwise determine the value of \(a\).
3. Show that the median value \(m\) of \(X\) satisfies the equation $$8 m ^ { 4 } - 28 m ^ { 2 } + 9 m - 4 = 0 .$$
4. Verify that the median value of \(X\) is 1.74, correct to \(\mathbf { 2 }\) decimal places.
5. Find \(\mathrm { E } ( X )\).
6. Determine the mode of \(X\).

5 An insurance company models the claims it pays out in pounds \(( \pounds )\) with a random variable \(X\) which has probability density function $$f ( x ) = \begin{cases} \frac { k } { x } & 1 < x < a
0 & \text { otherwise } \end{cases}$$ 5

1. The median claim is \(\pounds 200\)
   Show that \(k = \frac { 1 } { 2 \ln 200 }\)
   5
2. Find \(\mathrm { P } ( X < 2000 )\), giving your answer to three significant figures.
   5
3. The insurance company finds that the maximum possible claim is \(\pounds 2000\) and they decide to refine their probability density function. Suggest how this could be done.