A bakery sells rolls in units of a dozen.The demand X (in 1000 units) for rolls has a gamma distribution with parameters α = 3, θ = 0.5, where θ is in units of days per 1000 units of rolls. It costs 2 dollars to make a unit that sells for 5 dollars on the first day when the rolls are fresh. Any leftover units are sold on second day for 1 dollars.

let w denote the number of units to be made, then the above information suggests that the net profit u(x) as a function of the demand x (in units of days per 1000 units of rolls) can be modeled by $u(x)$=$\frac{5000}{x}$+$(w-\frac{1000}{x})-2w$ where we conveniently ignored the case 1000/x ≥ w for simplicity. Compute E[u(X)] and determine the number of units that should be made to maximize the expected value of the net profit.

Given the Gamma distribution $\frac{1}{\Gamma (\alpha)}$ $x^{\alpha-1}$ $e^\frac{-x}{\theta}$, I get $4 x^2*e^{-2x}$.

Plugging into $E(u(x))$ = $\int_{0}^{\infty} u(x)*4 x^2*e^{-2x} $, how am I supposed to solve this? (Note that w is a constant)

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    $\begingroup$ Use the function given and integrate... stuff cancels nicely. $\endgroup$ – Sean Roberson Feb 15 at 2:55

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