1. * Give an example of a density function on Œ0; 1] by giving a formula such that the density is finite at zero, unbounded at one, has a unique minimum in the open interval .0; 1/, and such that the median is .5.
2. The time taken by a group of students to finish a quiz has the density function f .x/ D cx2.15 -x/; 0 --S x --S 15:
(a) Find the normalizing constant c that makes f .x/ a density function.
(b) What percentage of the students take more than ten minutes to finish the quiz? Less than five minutes?
(c) What is the mean time taken to finish the quiz?
3. * (A Mixed Distribution). Suppose the damage claims on a partic- ular type of insurance policy are uniformly distributed on Œ0; 5] (in thousands of dollars), but the maximum payout by the insurance company is 2500 dollars. Find the CDF and the expected value of the payout, and plot the CDF. What is unusual about this CDF?