Log-Likelihood Functions for Continuous Probability Distributions

Given a random sample $x_1, x_2, \ldots, x_n$ from a continuous distribution with pdf $f(x \mid \theta)$, the likelihood function is the product

$$L(\theta) = \prod_{i=1}^n f(x_i \mid \theta).$$

The log-likelihood function is the natural logarithm of the likelihood:

$$\ell(\theta) = \ln L(\theta) = \sum_{i=1}^n \ln f(x_i \mid \theta).$$

Since $\ln$ is strictly increasing, $L(\theta)$ and $\ell(\theta)$ are maximized at the same value of $\theta$. The logarithm converts the product into a sum, which is far easier to handle algebraically and to differentiate when locating a maximum.

The formula has the same shape as in the discrete case; the only change is that $f$ now denotes a pdf rather than a pmf. For instance, the exponential pdf $f(x \mid \lambda) = \lambda e^{-\lambda x}$ has logarithm

$$\ln f(x \mid \lambda) = \ln \lambda - \lambda x.$$