Likelihood Functions: Parameters & Probabilities

[Cinitiator]

As far as I know, the definition of likelihood functions is the probability of a given random variable result given some parameter (please correct me if I'm wrong). What kind of parameters are usually handled by likelihood functions? Population parameters? Statistical model parameters? Both?

 

[Stephen Tashi]

As far as I know, the definition of likelihood functions is the probability of a given random variable result given some parameter (please correct me if I'm wrong).

For a continuous random variable x with probability density f(x), a number such as f(a) isn't "the probability that x = a". ( For example the desnity of a random variable x uniformly distributed on the interval [0, 1/2] is f(x) = 2 and 2 isn't a possible value for the probability of an event.) The density can be used to approximate the probability that x is in a small interval around a particular value and in many situations, you can think of the density at f(a) as "the probability that x = a" in order to remember the correct formulas. But f(a) isn't actually "the probability that x = a".

The fact that a value of the denstiy function isn't an actual probability explains why the phrase "maximum liklihood" is used instead of the simpler phrase "maximum probability".