A store sells a specific type of goods. The purchase quantity (่ฟ่ดง้) per week ๐ andcustomersโ demand on the goods per week ๐ are independent random variables, bothfollow ๐(10,20). Suppose that the profit of the store obtained by selling one unit of goodsis 1000 yuan. iowever, if the demand exceeds the purchase quantity, the store could ordergoods from other stores, and the profit per unit of goods is 500 yuan in this case. Computethe expected profit per week of the store by selling this type of goods. (10 points
Question
A store sells a specific type of goods. The purchase quantity (่ฟ่ดง้) per week ๐ andcustomersโ demand on the goods per week ๐ are independent random variables, bothfollow ๐(10,20). Suppose that the profit of the store obtained by selling one unit of goodsis 1000 yuan. iowever, if the demand exceeds the purchase quantity, the store could ordergoods from other stores, and the profit per unit of goods is 500 yuan in this case. Computethe expected profit per week of the store by selling this type of goods. (10 points
Solution
To solve this problem, we need to consider two scenarios: when the demand is less than or equal to the purchase quantity, and when the demand exceeds the purchase quantity.
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When the demand (Y) is less than or equal to the purchase quantity (X), the profit is 1000 yuan per unit. The probability density function (pdf) of a uniform distribution is 1/(b-a), where a and b are the lower and upper limits of the distribution. So, the expected profit E1 in this case can be calculated by integrating the product of the profit and the joint pdf of X and Y over the region where Y <= X. Since X and Y are independent, the joint pdf is the product of the individual pdfs.
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When the demand (Y) exceeds the purchase quantity (X), the profit is 500 yuan per unit for the excess demand. The expected profit E2 in this case can be calculated by integrating the product of the profit and the joint pdf of X and Y over the region where Y > X.
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The total expected profit E is the sum of E1 and E2.
Note: The actual computation of the integrals may require some mathematical skills and could be quite complex. It's recommended to use a software tool like MATLAB or Mathematica to do the computation.
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