1. ## Expected Profit

A gas station owner has a daily demand of 100X gallons of gas. (Note that X = .5 represents a demand of 50 gallons.) The distribution of daily demand is as follows:

$\displaystyle \displaystyle f(x)=\begin{cases}\frac{3}{2}\left(1-x+\frac{x^2}{4}\right), \ 0\leq x\leq 2\\0, \ \text{otherwise}\end{cases}$

The owner's profit is $10 for each 100 gallons sold if$\displaystyle X\leq 1$, and$15 per 100 gallons if $\displaystyle X>1$. Find the retailer's expected profit for any given day.

I know I am looking for

$\displaystyle \displaystyle\int_0^2 xf(x)dx$

How do I incorporate the profit into this integral?

2. Originally Posted by dwsmith
A gas station owner has a daily demand of 100X gallons of gas. (Note that X = .5 represents a demand of 50 gallons.) The distribution of daily demand is as follows:

$\displaystyle \displaystyle f(x)=\begin{cases}\frac{3}{2}\left(1-x+\frac{x^2}{4}\right), \ 0\leq x\leq 2\\0, \ \text{otherwise}\end{cases}$

The owner's profit is $10 for each 100 gallons sold if$\displaystyle X\leq 1$, and$15 per 100 gallons if $\displaystyle X>1$. Find the retailer's expected profit for any given day.

I know I am looking for

$\displaystyle \displaystyle\int_0^2 xf(x)dx$

How do I incorporate the profit into this integral?
E(X) = 1/2 therefore E(daily demand) = 100(1/2) = 50 therefore E(Profit) = (50)(10) = \$500.