2017 NYJC P2 Q5

From past records, the number of days of hospitalization for an individual with minor ailment can be modelled by a discrete random variable with probability density function given by \(\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix} \frac{6-x}{15},\mathrm{ for }x=1,2,3,4,5 \\ 0,\mathrm{ otherwise}\mathrm{.} \end{matrix} \right.\)

An insurance policy pays \(\$100\) per day for up to \(3\) days of hospitalization and \(\$25\) per day of hospitalization thereafter.

1. Calculate the expected payment for hospitalization for an individual under this policy.[4]
2. The insurance company will incur a loss if the total payout for \(100\) hospitalisation claims under this policy exceed \(\$24000\). Using a suitable approximation, estimate the probability that the insurance company will incur a loss for \(100\) such claims.[4]