Conditional expectation

2 artículos

Probability & Statistics GradoExplicación9 min

Zero Profit at Every Bid, Because Winning Says the Firm Was Cheap

Acceptance restricts the value to below your bid, where a uniform variable averages half of it, and doubling half your bid returns exactly your bid. The expected profit is therefore identically zero at every bid up to 100 and 100 minus b above it, so there is no optimal bid to find. With a general multiplier the profit is b squared times k minus 2, over 200, making doubling the exact break-even multiple, and the article shows a value distribution starting at 50 where the same bidder profits.

Recreational, History & Meta GradoExplicación10 min

The 25% Gain From Swapping Needs a Distribution That Does Not Exist

Requiring the fifty-fifty at every amount you could open forces the weights to satisfy f(x) = f(x/2)/2, whose only solutions are proportional to 1/x, and that integrates to infinity at both ends. Conditional on the pair, the swap gains the smaller amount or loses it with equal chance, which is zero and needs no assumption at all. The article carries a proper spread where the conditional answer is genuinely x/2, and the infinite-mean spread where swapping really is right at every observable amount.