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
- The median claim is \(\pounds 200\)
Show that \(k = \frac { 1 } { 2 \ln 200 }\)
5 - Find \(\mathrm { P } ( X < 2000 )\), giving your answer to three significant figures.
5 - 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.