Which of the following statements is true for both the binomial and Poisson distributions?

• A. Both distributions are only defined for continuous random variables.
• B. Both distributions require the number of trials to be infinite.
• C. Both distributions require the probability of success to be less than $0.5$.
• D. Both distributions describe the probability of a given number of events occurring in a fixed interval.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the nature of the binomial and Poisson distributions. Both are discrete probability distributions, meaning they describe probabilities for countable outcomes (like the number of successes or events), not continuous variables.
2. Step 2: Recall the binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. The number of trials is finite and fixed, not infinite.
3. Step 3: Recall the Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming these events happen independently and at a constant average rate. It does not require a fixed number of trials but rather focuses on the count of events in that interval.
4. Step 4: Note that neither distribution requires the probability of success to be less than 0.5. The binomial distribution can handle any probability between 0 and 1, and the Poisson distribution is parameterized by the average rate of events, not a probability of success.
5. Step 5: Conclude that the true statement common to both distributions is that they describe the probability of a given number of events occurring in a fixed interval (for Poisson) or fixed number of trials (for binomial), emphasizing their role in modeling counts of events.