The distribution of sum of random variables and the quotient (ratio) of two random variables are of interest in many areas of sciences. The study of these random variables and the problem of determining their exact probability density function, cumulative distribution function and other statistical parameters were examined by many authors for decades, for their importance in describing an application model. However, some problems were solved and other problems were examined and only approximations were given, but many still unsolved. In our work, we considered some of these problems, and our objective is to determine an exact closed expressions of the probability density function, cumulative distribution function and other statistical parameters. The problem was solved completely for the following sums and ratio: The sum of independent Exponential Random Variables (the Hypoexponential Random Variable) The sum of two independent Erlang Random Variables The sum of two independent Hyper-Erlang Random Variables The ratio of two independent Hypoexponential Random Variables The ratio of two independent Hyper-Erlang Random Variables Moreover, we give some properties of the expressions obtained. Also, for each topic, we give an example to illustrate our work.