Let's analyze the problem:

Problem Analysis

We are tasked to determine whether the given probability distribution is valid. A valid probability distribution must satisfy these two criteria:

1. Non-Negativity: All probabilities $P(x)$ must be greater than or equal to zero.
2. Normalization: The sum of all probabilities must equal 1.

The table is: $x: 1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7$ $P(x): -0.04 \quad 0.02 \quad 0.21 \quad 0.39 \quad 0.05 \quad 0.33 \quad 0.04$

Step 1: Check Non-Negativity

• $P(x) = -0.04$ for $x = 1$. This violates the non-negativity rule because probabilities cannot be negative.

Step 2: Check Normalization

Even though non-negativity is already violated, let’s check the sum of $P(x)$ for completeness: $\text{Sum of } P(x) = -0.04 + 0.02 + 0.21 + 0.39 + 0.05 + 0.33 + 0.04 = 1.00$ The sum equals 1, so the second condition is satisfied.

Conclusion

The distribution is not valid because it violates the non-negativity rule. Therefore, the correct answer is:

• D. The distribution is not valid. A probability cannot be negative.

Would you like more details on probability distributions?
Here are five related questions to expand your understanding:

1. How do you construct a valid probability distribution?
2. What happens if the sum of probabilities exceeds 1 or is less than 1?
3. Can probabilities ever exceed 1? Why or why not?
4. What is the difference between a probability distribution and a cumulative probability distribution?
5. How does a probability distribution apply to real-world situations?

Tip: Always verify both conditions (non-negativity and normalization) when validating a probability distribution.