Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a distribution \(F\) with continuous density \(f\). Prove that the histogram estimate with fixed grid points \(-\infty\max \left\{X_{1}, \ldots, X_{n}ight\}\) given by

\[\bar{f}_{n}(x)=\left(g_{i+1}-g_{i}ight)^{-1} n^{-1} \sum_{k=1}^{n} \delta\left\{X_{i}, ;\left(g_{i}, g_{i+1}ight]ight\}\]

is a valid density function, conditional on \(X_{1}, \ldots, X_{n}\).