The annual weather-related loss of an insurance company is modelled by a random variable \(X\) with probability density function\[f(x) = \left\{ {\begin{array}{*{20}{c}}
  {\frac{{2.5{{\left( {200} \right)}^{2.5}}}}{{{x^{3.5}}}},}&{x \geqslant 200} \\
  {0,}&{{\text{otherwise}}{\text{.}}}
\end{array}} \right.\]Find the median.

\(\int_{200}^M {\frac{{2.5{{\left( {200} \right)}^{2.5}}}}{{{x^{3.5}}}}{\text{d}}x}  = 0.5\)     M1A1A1

Note: Award M1 for the integral equal to \(0.5\)

   A1A1 for the correct limits.

\(\frac{{ - {{200}^{2.5}}}}{{{M^{2.5}}}}\left( {\frac{{ - {{200}^{2.5}}}}{{{{200}^{2.5}}}}} \right) = 0.5\)     M1A1A1

Note: Award M1 for correct integration

   A1A1 for correct substitutions.

\(\frac{{ - {{200}^{2.5}}}}{{{M^{2.5}}}} + 1 = 0.5 \Rightarrow {M^{2.5}} = 2{\left( {200} \right)^{2.5}}\)     (A1)

\(M = 264\)     A1

[8 marks]

Many students used incorrect limits to the integral, although many did correctly let the integral equal to \(0.5\).