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Previous work on non-parametric estimation has concerned three problems: (i) confidence intervals for an unknown quantile, (ii) population tolerance limits, (iii) confidence bands of an unknown cumulative distribution function $(cdf)$. For problem (iii) a solution has been available which is valid for any $cdf$ whatever, but for (i) and (ii) it has heretofore assumed that the population has continuous probability density. This paper validates the existing solutions of (i) and (ii) assuming only a continuous $cdf$. It then modifies these solutions so that they are valid for any $cdf$ whatever.