6 10 points total an insurance policy pays 1000 per day for up to 3 da

Question

6. (10 points total) An insurance policy pays $1000 per day for up to 3 days of hospitalization and$500 per day for each day of hospitalization thereafter. The number of days of hospitalization,X, is a discrete random variable with probability function p_{X}(k)=\left\{\begin{array}{cc}
\frac{6-k}{15} & \text { for } k=1,2,3,4,5 \\
0 & \text { otherwise }
\end{array}\right. (a) (5 points) Determine the expected payment for hospitalization under this policy. (b) (5 points) Determine the variance of the payment for hospitalization under this policy.