# Quick Problem with maximizing profit using different distributions

I need a little help with a problem I am working on. So here's the situation, a seller can produce a apple for $1.00$ and I need to find the optimal price of selling the apple and expected profit per buyer if the distribution of the value of the apple is EXP(1).

So I just need some confirmation on what I am doing. What I have is this:

Expected profit = $(v-1)\cdot (1-F(v))$

Optimal price = derivative of expected profit w.r.t. to $v$ set equal to zero then solve for $v$

I hope this makes sense guys. Like I said I just want make sure I am doing this right or else I am going to have to read the chapter for the 3rd time.

You're right. In general, if the valuation of a good follows the distribution $F(x)$, then the demand function is $D(x) = 1 - F(x)$ implying that the profit under a marginal cost $m$ is simply $$\Pi = (x-m)[1-F(x)]$$
Now, for the exponential distribution with parameter $\lambda$ a good trick to remember is that the optimal price equals $m+ \lambda^{-1}$, so your solution is $x=2$.