Final answer:
The method-of-moments estimator for θ in the uniform distribution is the sample mean divided by 1.5. For the Poisson distribution, the estimator for λ is the sample mean. The Poisson distribution can approximate the binomial distribution when the probability of success is small, and the number of trials is large.
Step-by-step explanation:
The subject question pertains to the derivation of a method-of-moments estimator for a parameter θ, based on a set of independent and identically distributed uniform random variables. To find the method-of-moments estimator for θ, we would use the fact that the mean of a uniform distribution on the interval (0, 3θ) is 1.5θ. Equating this to the sample mean, our method-of-moments estimator for θ would be the sample mean divided by 1.5.
For the Poisson distribution part, the second half of the student's question seems incomplete. However, the Poisson distribution is typically parameterized by λ (lambda), which represents the average rate (mean) of occurrences in the specified interval. If the student needs to estimate λ using the method of moments, they would equate the sample mean to λ since the mean of a Poisson distribution is λ.
In terms of choosing an appropriate distribution for a given problem, if the question contains a scenario where the probability of success is small and the number of trials is large, the Poisson distribution can be used as an approximation for the binomial distribution. However, if the scenario does not fit those criteria, one would typically use the binomial distribution directly.
The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size becomes larger, regardless of the distribution of the population. This is relevant for constructing confidence intervals for population means.