To determine whether the table represents a valid probability distribution, we must verify two conditions:

1. All probabilities $P(x)$ must be between 0 and 1 (inclusive):
   For each value in $P(x)$, check that $0 \leq P(x) \leq 1$.

2. The sum of all probabilities must equal 1:
   Calculate $\sum P(x)$ and ensure it equals 1.

Step 1: Verify $0 \leq P(x) \leq 1$

From the table:

• $P(0) = 0.2$, which is between 0 and 1.
• $P(1) = 0.05$, which is between 0 and 1.
• $P(2) = 0.15$, which is between 0 and 1.
• $P(3) = 0.55$, which is between 0 and 1.

All values satisfy this condition.

Step 2: Check if $\sum P(x) = 1$

$\sum P(x) = 0.2 + 0.05 + 0.15 + 0.55 = 0.95$

The total is 0.95, which does not equal 1.

Conclusion:

Since the probabilities do not sum to 1, the table does not represent a valid probability distribution.

Answer: No, it is not a probability distribution.

Would you like further details about this or related concepts? Here are some questions to deepen your understanding:

1. What happens if the sum of probabilities is greater than 1?
2. How could we adjust this table to make it a valid probability distribution?
3. Why is it important for probabilities to sum to exactly 1?
4. Can you identify real-world examples of valid probability distributions?
5. What is the role of normalization in probability distributions?

Tip: Always check both conditions (sum and range) to confirm a valid probability distribution!