Likelihood Function

Definition

Let

• $X_1,\dots ,X_n$ : Random sample
• $x_1,\dots ,x_n$ : Realization of $X_1,\dots ,X_n$
• $\theta$ : Parameter
• $f(x;\theta )$ : pdf of $X_1,\dots ,X_n$

If

$$\begin{array}{rl}L(\theta )& =L(\theta ;x_1,\dots ,x_n)\\ & =\prod _{i=1}^{n}f(x_i;\theta )\end{array}$$

Then we say $L(\theta )$ is the likelihood function of $X_1,\dots ,X_n$

About likelihood function

Likelihood function returns the probability of observed data (the $x$‘s) expressed as a function of the parameter ($\theta$).

See also

• Maximum Likelihood Estimation (MLE)

Related definitions

• Maximum Likelihood Estimator

Related theorems