I am doing a study problem and I would like to know if my answers look fine.

$X$ is a random variable that represents the demand for a product with the following density function: $$f(x) = \frac1{e^x} ,\ x > 0$$

$c > 0$ is the number of units in stock

$ax$ are the earnings for $x$ units sold

$b(c-x)$ is the loss caused by the unsold units

1) Find the earnings as a function of $X$ and $c$

My answer: $$W = aX - b(c - X)$$

2) What are the expected earnings?

To get this answer I assumed that $X$ is a continuous random variable. My answer: $$E[X] = 1$$ $$E[W] = a - b(-1 + c)$$

3) What is the value of $c$ that maximizes the expected earnings?

My answer: $$c = E[X]$$

  • $\begingroup$ Isn´t $\mathbb E(aX-bc+bX)=a-bc\color{red}+b$? The rest looks fine. $\endgroup$ – callculus May 23 at 20:24
  • 1
    $\begingroup$ @callculus Thank you, you are right. $\endgroup$ – Lollipop May 23 at 20:51
  • $\begingroup$ You´re welcome. In general it was a good job. $\endgroup$ – callculus May 23 at 20:55

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