Derivation of Maximum Likelihood Estimators (MLEs) for Simple Distributions

Students frequently confuse the Likelihood function

L(\theta)

with a probability density function. Crucially,

L(\theta)

is a function of

\theta

with fixed

x

, not a distribution over

x

. Thus, it does not necessarily integrate to one, and the 'area' under

L(\theta)

lacks a standard probabilistic interpretation.

Institutional Reference: Applied Statistics