Very confused about this seemingly simple probability question

[JasonJo]

you have $1000 and a certain commodity costs $2 an ounce. Suppose that after 1 week, there is a 50% that the commodity will cost $1 and a 50% that the commodity will cost $4.

i already know how to do the expected value of cash, but

(b) If your objective is to maximize the expected amount of commodity that you possesses at the end of the week, what strategy should you employ?

my professor said setup a random variable Y. but he said the random variable Y represents the amount of commodity i buy today, but means, just buy 500 ounces of the commodity to maximize it.

he also hinted that this answer will involve E(g(x))

any help or helpful hints?

so Y = {250, 500, 1000}
but i don't understand how g(x) or how you calculate P(y=Y)

 

[StatusX]

I'm not sure, but it looks like to maximize the amount of the commodity you possess, you shouldn't buy all of it at the beginning. For example, if you waited until the end of the week and then spent all of your money to buy as much as you could, there's a 50% chance you'll get 1000 and a 50% chance you'll get 250, for an expected amount of 525, better than the 500 you get from buying first. I don't know if this is the best you can do, though.

 

[JasonJo]

yeah i got E(Y) = 625 (500 + 125)

but, wht about this?

you buy some when it is $2 per ounce and then you wait till the price changes and then buy more?

ugh, he said it was easy lol