Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a discrete distribution with distribution function \(F\) and probability distribution function \(f\). Assume that \(F\) is a step function with steps at points contained in the countable set \(D\). Consider estimating the probability distribution function as

\[\hat{f}_{n}(x)=\hat{F}_{n}(x)-\hat{F}_{n}(x-)=n^{-1} \sum_{k=1}^{n} \delta\left\{X_{k} ;\{x\}ight\}\]

for all \(x \in \mathbb{R}\). Prove that \(\hat{f}_{n}(x)\) is an unbiased and consistent estimator of \(f(x)\) for each point \(x \in \mathbb{R}\).