Question

Exhibit 6-10a. You are asked to bid on a first edition copy of Snedecor's Statistical Methods at an auction. You are bidding for a friend and the friend gives you $11600 and tells you that if you win the item for her then you can keep whatever is left over and if you lose the auction then you simply return the $11600 to her. At the auction there is only one other bidder and you believe their bid for the item will be uniformly distributed between $2300 and $10100. The winner of the auction is the highest bidder and they pay the amount of the loser's bid (similar to eBay).

[R] Refer to Exhibit 6-10a. If you submit a bid of $8140 what is your expected profit, E(profit)? Note: profit is revenue minus cost.

Answer 1

Exhibit 6-10a

At the auction there is only one other bidder and you believe their bid for the item will be uniformly distributed between $2300 and $10100.

X : Bid amount by only other bidder

X is uniformly distributed between $2300 and $10100.

Probability mass function X :

And Cumulative Distribution function

Z : Your profit

You win the item if you bid of $8140 is higher than the only other bidder , then your profit : Z = 11600 - 8140= 3460

Probability that you bid of $8140 is higher than the only other bidder, or the only other bidder bid is less than $8140

i.eP(Z=3460) = P(X8140) = F(8140)

You win the item if you bid of $8140 is less than the only other bidder , then your profit : Z = 11600 - 11600= 0

Probability that you bid of $8140 is lower than the only other bidder, or the only other bidder bid is higher than $8140

P(Z=0)= P(X>8140) = 1- P(X8140) = 1-0.7487 = 0.2513

Z : Profit

P(Z=3460) = 0.7487

P(Z=0) = 0.2513

Expected Profit = $2590.50