Known-rate Laplace location confidence interval ($\overline{X}\pm z_{1-\alpha /2}\sqrt{2}/(\lambda \sqrt{n})$)

For an independent and identically distributed Laplace distribution sample with density $\lambda e^{-\lambda |x-\mu |}/2$ and known rate $\lambda >0$, the sample mean has mean $\mu$ and variance $2/(n{\lambda }^2)$. The central limit theorem gives the displayed asymptotic confidence interval for $\mu$ at confidence level $1-\alpha$. The sample mean need not be a maximum-likelihood estimator for this construction to be valid. The Laplace location and scale maximum likelihood location is a sample median; that is a different choice of statistic.