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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49 Answers

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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