I have this problem which I have been struggling with now for a while.
"You bid on an object at a silent auction. You know that you can sell it later for $100 and you estimate that the maximum bid from others is uniform on [70,130]. (For convenience, you assume that it is continuous, thus disregarding the possibility of two equal bids). How much should you bid to maximize your expected profit, and what is the maximum of the expected profit?"
My attempt to solve this goes like this.
Let the random variable $Y$ denote my profit after I sell the object, and let
$X$ be the random variable denoting the other peoples maximum bid.
We know from the text that $X$∽$unif[70,130]$ and we can therefore easily compute $X$'s pdf and expected value.
Now, my plan is to find a function $g$ such that $Y=g(X)$ and then calculate $E[Y]=E[g(X)]$ and later maximize. I tried to set $Y=100-X$ but that doesnt make any sense. I have no idea how to continue from here. I would really appreciate a nudge in the right direction on this one.
Any idea? Thanks!
Pretty clearly, your optimal bid should be between $70$ and $100$. If you bid $x$ within this range, your chance of winning is $\frac{x-70}{60}$, and if you win, your resale profit is $100-x$.
So your expected value for bidding $x$ in the range $[70,100]$ is $\frac{(x-70)(100-x)}{60}$.
You should be able to maximize that expression with respect to $x$.