Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 9 / b / ii / Solution

By independence, the likelihood function is the product of the individual probability density functions:Writing $A(\mu )={\sum }_i|x_i-\mu |$, the log-likelihood function is $\ell =n\ln \lambda -n\ln 2-\lambda A(\mu )$. Here $\lambda$ is a rate, the reciprocal of the scale in the usual Laplace distribution notation.