Question

Suppose you are interested in bidding on a parcel of land and you know that one other bidder is interested. The seller has announced that the highest bid will be accepted. The other competitor’s bidding price for the land will vary evenly from $72,000 to $85,500.

1. Let B be the competitor’s bid for the parcel of land. What are the distribution and parameter(s) of B?

2. What is expected value and standard deviation of the competitor’s bid?

3. What is the probability that the competitor’s bid will be less than $80,000?

4. What values mark the middle 50% of the competitor’s bids?

5. You and the competitor each put in a sealed bid (i.e. you do not know the competitor’s bid and he does not know yours). Suppose that you bid $79,000. What is the probability that your bid will be accepted? (i.e. that yours will be higher than the competitor.)

6. The competitor has hinted that he will not go higher than $82,750. What is the probability that his actual bid is above $80,200?

Answer 1

Suppose you are interested in bidding on a parcel of land and you know that one other bidder is interested. The seller has announced that the highest bid will be accepted. The other competitor’s bidding price for the land will vary evenly from $72,000 to $85,500.

so if  B be the competitor’s bid for the parcel of land. Then,

f(B) = 1/(85500 - 72000) = 1/13500 ;  $72,000 < B < $85,500

Cumulative probability distribution

F(B) = (x - 72000)/13500 ;   $72,000 < B < $85,500

Here the distribution is Uniform Distribution and parameter a = 72000, b = 85500

(b) Expected value = E[B] = (85500 +72000)/2 = $ 78,750

standard deviation = SD[B] = sqrt [(b-a)²/12] = sqrt [(85500 - 72000)²/12] = $ 3897.11

(c) P(B < $ 80000) = (80000 - 72000)/13500 = 0.5926

(d) Here let say middle 50% values are B₁ and B₂

so as this is symmetric

P(B < B₁) = 0.25 = P(B > B₂)

so here

B₁ = 72000 + 0.25 * 13500 = $ 75375

B₂ = 85500 - 0.25 * 13500 = $ 82125

so here 50% of values are in between $ 75375 and $ 82125.

(e) Here our bid is $ 79000.

Our bid would be accepted if cometitors's bid is greater than $ 79,000.

so,

P(Our bid will be accepted) = P(B > $ 79,000) = 1 - P(B < $ 79,000)

= 1 - (79000 - 72000)/13500

= 1 - 0.5185 = 0.4815

(f) Here as it is said that he will not go higher than $82,750, writing mathematically,

P(B > $ 80,200 l B < $ 82750) = P($ 80200 < B < $ 82750)/ P(B < $ 82750)

P($ 80200 < B < $ 82750) = (82750 - 72000)/13500 - (80200 - 72000)/13500 = 0.1889

P(B < $ 82750) = (82750 - 72000)/13500 = 0.7963

P(B > $ 80,200 l B < $ 82750) = 0.1889/0.7963 = 0.2372