Question

It is known that the content of milk that is actually in a bag distributes normally with an average of 900 grams and variance 25 square grams. Suppose that the cost in pesos of a bag of milk is given by 𝐶(𝑥) = { 3800 𝑠𝑖 𝑥 ≤ 890 4500 𝑠𝑖 𝑥 > 890 Find the expected cost.

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Answer to a math question It is known that the content of milk that is actually in a bag distributes normally with an average of 900 grams and variance 25 square grams. Suppose that the cost in pesos of a bag of milk is given by 𝐶(𝑥) = { 3800 𝑠𝑖 𝑥 ≤ 890 4500 𝑠𝑖 𝑥 > 890 Find the expected cost.

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Esmeralda
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To find the expected cost, we need to calculate the weighted average of the cost function, considering the probabilities of each case. Given: Average of milk content = 900 grams Variance of milk content = 25 square grams Let's calculate the probabilities for each case: Case 1: x ≤ 890 This corresponds to the lower range of milk content. To find the probability, we can use the cumulative distribution function (CDF) of the normal distribution. P(x ≤ 890) = CDF(890, 900, √25) Case 2: x > 890 This corresponds to the upper range of milk content. P(x > 890) = 1 - P(x ≤ 890) Now, let's calculate the probabilities: P(x ≤ 890) = CDF(890, 900, √25) = CDF(-2, 0, 5) ≈ 0.0228 P(x > 890) = 1 - P(x ≤ 890) = 1 - 0.0228 ≈ 0.9772 Next, let's calculate the expected cost: Expected cost = P(x ≤ 890) * Cost for x ≤ 890 + P(x > 890) * Cost for x > 890 Expected cost = 0.0228 * 3800 + 0.9772 * 4500 Expected cost ≈ 86.76 + 4397.4 Therefore, the expected cost is approximately: **Expected cost ≈ 4484.16 pesos**

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