In statistics and probability theory, one the most important statistics is the sums of random variables. After introducing a probability distribution, determining the sum of n independent and identically distributed random variables is one of the interesting topics for authors. This paper presented the probability density function for the sum of n independent and identically distributed random variables such as Shanker, Akash, Ishita, Rani, Pranav and Ram Awadh. In order to determine all aforementioned distributions, the problem-solving methods are applied which is based on the change-of-variables technique. The mth moments for them were also accurately calculated. Besides, the reliability and the mean time to failure of a 1 out of n cold standby spare system has also been evaluated under the Lindley components failure time.