1. **Problem Statement:** Determine whether the given distribution is a valid probability distribution for a discrete random variable $X$ with values $5, 6, 7, 4$ and probabilities $0.22, 0.35, 0.18, 0.35$ respectively. 2. **Formula and Rules:** A probability distribution for a discrete random variable must satisfy two conditions: - Each probability $P(X=x)$ must be between 0 and 1 inclusive. - The sum of all probabilities must equal 1. 3. **Check each probability:** - $0.22$ is between 0 and 1. - $0.35$ is between 0 and 1. - $0.18$ is between 0 and 1. - $0.35$ is between 0 and 1. All probabilities satisfy the first condition. 4. **Sum the probabilities:** $$0.22 + 0.35 + 0.18 + 0.35 = 1.10$$ 5. **Interpretation:** The sum of probabilities is $1.10$, which is greater than 1. This violates the second condition for a valid probability distribution. 6. **Conclusion:** Since the sum of probabilities is not equal to 1, the given distribution is **not** a valid probability distribution for the discrete random variable $X$.