Question: A filter for a certain water purifier has a life (in days) with a probability density function (pdf) given by

\[f\left( x \right)=\left\{ \begin{aligned} & 0\text{ }\,\,\,\,\text{ for }x<10 \\ & \frac{200}{{{x}^{3}}}\text{ for }x>10 \\ \end{aligned} \right.\]

Find the expected lifetime of such a filter, in days. (1.5 marks)