Likelihood functions

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

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".