13, 15, 17, 19
1, 2, 3, 10
Let \(X \sim \text{Binomial}(n, \theta)\), and consider the estimator \(\bar{\theta}_{n} = \dfrac{X + 1}{n + 2}\) for the success probability \(\theta\).
a. Determine the expected value of \(\bar{\theta}_{n}\). What is the expected value of \(\bar{\theta}_{n}\) as \(n \to \infty\)?
b. Determine the variance of \(\bar{\theta}_{n}\).
c. Determine the standard error of \(\bar{\theta}_{n}\).
d. How does the estimator \(\bar{\theta}_{n}\) compare the estimator \(\widehat{\theta}_{n} = X /n\) we discussed in class?