Likelihood Function

Definition

With $X_1,\dots ,X_n$ as a Joint Probability Mass Function or Joint PDF With observed samples $X_1=x_1,\dots ,X_n=x_n$ The likelihood function $L(\theta |x_1,\dots ,x_n)=f(x_1,\dots ,x_n|\theta )$ Can be calculated as: $L(\theta )={\Pi }_{i=1}^n{f}_{\theta }(x_1)$

Example

For Bernoulli Trials $0<\theta <1$ We flip coins, and we observe IID sample $(1,0,1,1,0)$ $L(\theta |X_1=1,X_2=0,X_3=1,X_4=1,X_5=0)$ $={\theta }^3(1-\theta {)}^2$ Note that this is the probability of getting the sample $(1,0,1,1,0)$.