Distribution of continuous random variable of one and two dimentions

In present teaching materials of Theory of possibility and mathematical statistics, the statement of probability distribution of 1-D and 2-D random variables is not succinct and systematic. We derive probability density function of random variables without using an integral representation. As first step, for 1-D monotone function of random variable, the method of taking derivatives of composite functions is used. Then, for non-monotone function, we divide the interval of variables into several so-called monotone regions. In this case probability density is given. To 2-D case, the method is the same, and the only difference is that the variable transformation is a little bit complicated.