This paper is mainly connected with the approximation properties of the positive linear operators which are representative expectation of some random variables. Firstly, it was introduced the generalizations of Korovkin type approximation theorem via the probabilistic methods. Also, it was computed the rates of convergence by means of modulus of continuity and moments of some random variables. Finally, some important results of the approximation theory were obtained; one of these important results is the difference between using appropriate operator and function and function's derivatives. A medical application performed on hypothetical data to the statistical significance of the findings has been mentioned.