Probability and Likelihood

given some distribution, there is an associated probability of observing some value $$x_i$$ on the random variable $$X$$

given some observations $$x_1, x_2$$, there is some likelihood that there is some underlying distribution with parameters $$\theta$$

"Point estimation is the process of obtaining an estimate of a parameter theta from an observed sample of data x sub 1 comma x sub 2 comma ellipsis comma x sub n. This is achieved via a point estimator (or estimator for short), theta hat, [theta hat is pronounced ‘theta hat’.] which is a function of the random variables uppercase X sub 1 comma uppercase X sub 2 comma ellipsis comma uppercase X sub n underlying the data, but not of theta.

The realisation of the estimator for a particular sample x sub 1 comma x sub 2 comma ellipsis comma x sub n is a point estimate (or just an estimate). This is a numerical value (and not a random variable), but is also often denoted theta hat. " @todo M347