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 \(f\). Let \(-\infty\max \left\{X_{1}, \ldots, X_{n}ight\}\). Consider the estimate of \(F\) given by

\[\bar{F}_{n}(x)=\hat{F}_{n}\left(g_{i}ight)+\left(x-g_{i}ight)\left[\hat{F}_{n}\left(g_{i+1}ight)-\hat{F}_{n}\left(g_{i}ight)ight]\]

when \(x \in\left[g_{i}, g_{i+1}ight]\). Prove that this estimate is a valid distribution function conditional on \(X_{1}, \ldots, X_{n}\).