Refer to Exercise 3.64 . The maximum likelihood estimator...

Refer to Exercise $3.64 .$ The maximum likelihood estimator for $p$ is $Y / n$ (note that $Y$ is the binomial random variable, not a particular value of it). a. Derive $E(Y / n)$. In Chapter 9 , we will see that this result implies that $Y / n$ is an unbiased estimator for $p.$ b. Derive $V(Y / n) .$ What happens to $V(Y / n)$ as $n$ gets large?

Refer to Exercise $3.64 .$ The maximum likelihood estimator for $p$ is $Y / n$ (note that $Y$ is the binomial random variable, not a particular value of it).
a. Derive $E(Y / n)$. In Chapter 9 , we will see that this result implies that $Y / n$ is an unbiased estimator for $p.$
b. Derive $V(Y / n) .$ What happens to $V(Y / n)$ as $n$ gets large?

Key Concepts

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model by maximizing the likelihood function, which measures how likely it is to observe the given data under various parameter values. In this context, the estimator for p, given as Y/n, is derived by finding the parameter value that makes the observed data most probable under a binomial model.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability p of success. It is characterized by its parameters n (number of trials) and p (probability of success), and has key properties such as an expected value of np and a variance of np(1-p), which are fundamental in the analysis of estimators like Y/n.

Unbiased Estimator

An unbiased estimator is a statistical estimator whose expected value is equal to the parameter it estimates. In this situation, showing that E(Y/n) equals p demonstrates that Y/n is an unbiased estimator for p, meaning that, on average, it correctly estimates the true parameter.

Variance and Asymptotic Behavior

Variance measures the spread or dispersion of an estimator's sampling distribution. For the estimator Y/n, its variance is derived as a scaled version of the binomial variance, specifically p(1-p)/n. As n increases, the variance decreases, illustrating that the estimator becomes more precise with larger sample sizes. This asymptotic behavior is important in large-sample theory and for understanding the efficiency of estimators.

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