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Math Help - Expected Profit

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    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 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 X\leq 1, and $15 per 100 gallons if X>1. Find the retailer's expected profit for any given day.

    I know I am looking for

    \displaystyle\int_0^2 xf(x)dx

    How do I incorporate the profit into this integral?
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    Quote Originally Posted by dwsmith View Post
    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 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 X\leq 1, and $15 per 100 gallons if X>1. Find the retailer's expected profit for any given day.

    I know I am looking for

    \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.
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