Estimators

Unbiased Estimator

Consider iid random variables \(X_1, X_2, \ldots, X_n \sim \text{Uniform}(N)\) where each random variable can take the values \(1,2,\ldots,N\). Notice that

\[ \mathbb E[X_k] = \sum_{i=1}^n i = \frac{N+1}{2} \]

Let the sample mean be,

\[ \bar X \equiv \frac{1}{n}\sum_{i=1}^n X_i \]

The expectation value of the sample mean is

\[ \mathbb E[\bar X] = \frac{N+1}{2} \]