Maximum likelihood estimator
Let us consider the discrete probability distributions as seen in the Discrete distributions section of Chapter 1, Data Characteristics. We saw that a binomial distribution is characterized by the parameters in n and p, the poisson distribution by 
, and so on. Here, the parameters completely determine the probabilities of the x values. However, when the parameters are unknown, which is the case in almost all practical problems, we collect data for the random experiment and try to infer about the parameters. This is essentially inductive reasoning and the subject of statistics is essentially inductive driven as opposed to the deductive reasoning of mathematics. This forms the core difference between the two beautiful subjects. Assume that we have n observations X1, X2,…, Xn from an unknown probability distribution 
, where 
may be a scalar or a vector whose values are not known. Let us consider a few important definitions that form the core of statistical inference...
