if (th tis differentiable function with respect to th and the maximum is attained in the interior of thand there exists unique maximum pointthen we can find the mle of th by solving the equations (th thi example (bernoulli random samplesuppose we have data tn { xn and assume the model xn ~iid ber(ththenthe likelihood function is given by (th tn = th xi ( th) -xi th ( th) - th ( where : xn =nx the log-likelihood is (ths ln th ( sln( ththrough differentiation with respect to thwe find the derivative - th th th( th th ( solving (th gives the ml estimate th and ml estimator th confidence intervals an essential part in any estimation procedure is to provide an assessment of the accuracy of the estimate indeedwithout information on its accuracy the estimate itself would be meaningless confidence intervals (also called interval estimatesprovide precise way of describing the uncertainty in the estimate let xn be random variables with joint distribution depending on parameter th th let be statisticsthat ist ( xn ) are functions of the databut not of th the random interval ( is called stochastic confidence interval for th with confidence if pth [ for all th th interval estimates stochastic confidence interval ( if and are the observed values of and then the interval ( is called the (numericalconfidence interval for th with confidence for every th th if the right-hand side of ( is merely heuristic estimate or approximation of the true probabilitythen the resulting interval is called an approximate confidence interval the probability pth [ th is called the coverage probability for confidence intervalit must be at least for multidimensional parameters th rd the stochastic confidence interval is replaced with stochastic confidence region rd such that pth [th for all th (numericalconfidence interval coverage probability confidence region