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To calculate \( \mu_X \) and \( \mu_Y \), we need to find the expected values for \( X \) and \( Y \), which represent the average number of weeks for each step based on their probability distributions.

### Step 1: Calculate \( \mu_X \)
The expected value \( \mu_X \) is calculated using:
\[
\mu_X = \sum X \cdot P(X)
\]
Using the values from the table:
- For \( X = 1 \): \( P(X=1) = 0.3 \)
- For \( X = 2 \): \( P(X=2) = 0.5 \)
- For \( X = 3 \): \( P(X=3) = 0.2 \)

So,
\[
\mu_X = (1 \cdot 0.3) + (2 \cdot 0.5) + (3 \cdot 0.2) = 0.3 + 1.0 + 0.6 = 1.9
\]

Thus, \( \mu_X = 1.9 \).

### Step 2: Calculate \( \mu_Y \)
Similarly, the expected value \( \mu_Y \) is calculated as:
\[
\mu_Y = \sum Y \cdot P(Y)
\]
Using the values from the table:
- For \( Y = 1 \): \( P(Y=1) = 0.1 \)
- For \( Y = 2 \): \( P(Y=2) = 0.15 \)
- For \( Y = 3 \): \( P(Y=3) = 0.4 \)
- For \( Y = 4 \): \( P(Y=4) = 0.3 \)
- For \( Y = 5 \): \( P(Y=5) = 0.05 \)

So,
\[
\mu_Y = (1 \cdot 0.1) + (2 \cdot 0.15) + (3 \cdot 0.4) + (4 \cdot 0.3) + (5 \cdot 0.05) = 0.1 + 0.3 + 1.2 + 1.2 + 0.25 = 3.05
\]

Thus, \( \mu_Y = 3.05 \).

### Final Answers
- \( \mu_X = 1.9 \)
- \( \mu_Y = 3.05 \)

Would you like more details or have any questions on these calculations?

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Here are some additional questions to consider:
1. What is the variance for \( X \) and \( Y \) in this scenario?
2. How would the expected time change if one of the probabilities increased?
3. What happens to \( \mu_Y \) if the probability for \( Y = 3 \) becomes 0.5?
4. How can these expectations influence project planning?
5. If we combine both steps, what is the total expected time for the process?

**Tip:** Expected value calculations are often the basis for project timelines and resource planning, as they provide a probable estimate for completion time.

Calculate the expected values μX and μY based on the provided probability distributions for X and Y. Round to two decimal places.

Learn how to calculate the expected values μX and μY based on specific probability distributions for two steps in a process. Includes detailed step-by-step calculations of expected values for X and Y. Suitable for advanced high school students studying probability and statistics.

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