Small point: If the vector X is drawn from e.g. the set of real real numbers, the likelihood function does not have to take values in [0,1]. As an example, consider the the case of the exponetial distribution over the set of non-negative real numbers. Clearly, if the variable x is distributed µ—exp(-µ—x) and X is a vector of realised values of x such that X= [0 0,01 0,02], then the likelihood function can take values larger than 1. E.g. if µ = 2, then the likelihood functiona app. takes the value 7,534.

This may seem a small point, but I often come across people who think that a contiuous probability distribution can only take values smaller than or equal to 1, when this is clearly nonsense.