Assume you sell chocolate bars for a price x greater than 0 and less than 10. You adjust the price often as you make a sell. After a while you construct a model measuring the sucess rate of making a sell with a price $x$ using the function $f(x)$.
Assuming the model is correct, what is the best price to charge such that you maximize your expected pay off value.
I am a bit confused about what I should do here exactly. I mean I could compute the expected pay off as $\int_0^{10} xf(x)dx$, but this value does not depend on $x$.
Best Answer
Let $f(x)$ denote the probability that if the price is $x$, then the sell occurs.
Let $t$ be the base price. Then, $x-t$ is the payoff if the sell occurs, and $-t$ is the payoff if the sell doesn't occur.
Then, the expected payoff at price $x$ is : $$ \begin{align} \text{payoff}(x) &=f(x)(x-t)+(1-f(x))(-t) \\ &= xf(x)-t \end{align}$$
Thus, $$\text{best price }= \displaystyle \text{arg max}_{0 \leq x \leq 10} xf(x) $$
This also holds if we say that there is no loss if the sell doesn't occur.