Question: User: P(X \leq 5) = P(X < 5) + P(X = 5) for only discrete distributions. for only continuous distributions. for all distributions. for some discrete distributions but not all.

1. **State the problem:** We want to understand for which types of probability distributions the equation $$P(X \leq 5) = P(X < 5) + P(X = 5)$$ holds true. 2. **Recall definitions:** - For a discrete random variable, probabilities at specific points can be positive, so $$P(X = 5)$$ can be nonzero. - For a continuous random variable, the probability at any single point is zero, so $$P(X = 5) = 0$$. 3. **Analyze the equation:** - The event $$X \leq 5$$ includes all outcomes where $$X < 5$$ plus the outcome where $$X = 5$$. - Therefore, the equation $$P(X \leq 5) = P(X < 5) + P(X = 5)$$ is a direct consequence of the additivity of probabilities for disjoint events. 4. **Apply to discrete distributions:** - Since discrete distributions can have positive probability at $$X=5$$, the equation holds exactly. 5. **Apply to continuous distributions:** - Since $$P(X=5) = 0$$ for continuous distributions, the equation reduces to $$P(X \leq 5) = P(X < 5) + 0 = P(X < 5)$$. - So the equation still holds, but $$P(X=5)$$ is zero. 6. **Conclusion:** - The equation $$P(X \leq 5) = P(X < 5) + P(X = 5)$$ holds for all distributions because it is a basic property of probability measures. **Final answer:** The equation holds for all distributions.