(a) In a closed auction, the buyers submit sealed bids so that no bidder knows the bids of the others. The highest bid wins. Suppose that the bids are independent r.v.’s uniformly distributed on [a,a+d]. For the case of n bidders, find the distribution and expectation of the price for which the item will be sold. (Advice: It is convenient to consider “surplus” values uniformly distributed on [0,d], and add a at the end.)

(b) (i) Do the same if the number of bidders is a Poisson r.v. N with parameter λ. (Advice: Condition on N.) (ii) Assuming λ to be integer, compare the mean value with that of part (a) for n = λ. (iii) may compare the result with that of part (a) in the sense of the first stochastic dominance.