Given the following table, does it represent a valid discrete probability distribution? x: $1$ $2$ $3$ $4$ P(x): $0.2$ $0.3$ $0.4$ $0.1$

• A. Yes, because all probabilities are between $0$ and $1$ and their sum is $1$.
• B. No, because there are negative probabilities.
• C. No, because the sum of the probabilities is less than $1$.
• D. No, because one of the probabilities is greater than $1$.

Answer: (A)

Verified step by step guidance

1. Recall the two main criteria for a valid discrete probability distribution: (1) each probability must be between 0 and 1 inclusive, and (2) the sum of all probabilities must equal 1.
2. Check each probability value given: 0.2, 0.3, 0.4, and 0.1. Verify that each is within the range 0 \leq P(x) \leq 1.
3. Calculate the sum of all the probabilities: 0.2 + 0.3 + 0.4 + 0.1. This sum should be exactly 1 for the distribution to be valid.
4. If all probabilities are between 0 and 1 and their sum equals 1, then the table represents a valid discrete probability distribution.
5. If any probability is negative or greater than 1, or if the sum of probabilities is not equal to 1, then the table does not represent a valid discrete probability distribution.