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?

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.