Math Help - Expected Profit

1. Expected Profit

Sorry, I already asked a question today, but it would be awesome if someone could help me with this one, I feel like I'm close:

The annual profit Y(in \$100,000) can be expressed as a continuous function of drug demand x(in 1,000): Y(x) = 2(1-e^(-2x)). Suppose the demand for their drug has the probability function: f(x)= 6e^(-6x), x>0. Find the company's expected annual profit.

So do I have to start by doing ? I'm sure I have the upper bound wrong...

2. The way I see this problem, it is a mixed distribution where the given profit function Y is conditional on X:

$f_{Y|X}(y|x)=2(1-e^{-2x})$

and therefore to find E(Y) you need first to find marginal distribution $f_Y(y)$ and then go from there.

But I may be totally wrong!

3. Hello,

$E[Y]=E[E[Y|X]]=E[E[2(1-e^{-2X})|X]]=E[2(1-e^{-2X})]$, which is the formula alakaboom1 wrote.
As for the boundaries, it should rather be on the whole set of real numbers.
Then since f(x)=0 if x<0, the boundaries will indeed go from 0 to infinity.

Volga : there's something disturbing in what you wrote. $2(1-e^{-2x})$ is not a pdf. We just have that $Y=2(1-e^{-2X})$, so in order to find Y's pdf, there's a change of variable to make in X's pdf

4. Sorry!