To find the expected number of complaints per day ($E[X]$), we use the formula for the expectation of a discrete random variable:

$E[X] = \sum (x \cdot P(x)),$

where $x$ represents the number of complaints per day and $P(x)$ is the associated probability.

Data from the table:

• $x = 0, 1, 2, 3, 4, 5, 6$
• $P(x) = 0.01, 0.08, 0.11, 0.28, 0.32, 0.12, 0.08$

Calculation:

$E[X] = (0 \cdot 0.01) + (1 \cdot 0.08) + (2 \cdot 0.11) + (3 \cdot 0.28) + (4 \cdot 0.32) + (5 \cdot 0.12) + (6 \cdot 0.08)$

$E[X] = 0 + 0.08 + 0.22 + 0.84 + 1.28 + 0.60 + 0.48$

$E[X] = 3.50$

Final Answer:

The expected number of complaints per day is 3.50.

Would you like a breakdown of these calculations or further clarification?

Related Questions:

1. How is the formula for expected value derived, and why does it work?
2. What is the significance of the expected value in this context?
3. Can the expected number of complaints ever be a non-integer, and what does that mean practically?
4. What would happen to $E[X]$ if probabilities were redistributed?
5. How do you calculate variance or standard deviation for this probability distribution?

Tip:

Always ensure that the probabilities sum up to 1 when working with discrete probability distributions to verify correctness.