Basic Concepts
For the exponential distribution, the pdf is
and so the likelihood function for a data set {x1, …, xn} is
Maximizing L(λ) is equivalent to maximizing the log-likelihood function LL(λ) = ln L(λ).
It turns out that LL is maximized when λ = 1/x̄, which is the same as the value that results from the method of moments (Distribution Fitting via Method of Moments). At this value, LL(λ) = n(ln λ – 1).
Reference
Wikipedia (2020) Exponential distribution
https://en.wikipedia.org/wiki/Exponential_distribution