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The maximum-likelihood estimate for the rate parameter $\lambda$ is derived by maximizing the likelihood function $\lambda^n \exp(-\lambda \sum x_i)$. The resulting estimate is $n / \sum x_i$, which matches the moment estimate.

Conditions: The observations are independent.; The observations follow an exponential distribution.; The rate parameter $\lambda$ is positive.

The maximum-likelihood estimate for the rate parameter $\lambda$ is derived by maximizing the likelihood function $\lambda^n \exp(-\lambda \sum x_i)$. The resulting estimate is $n / \sum x_i$, which matches the moment estimate.

Conditions: The observations are independent.; The observations follow an exponential distribution.; The rate parameter $\lambda$ is positive.