Further theories on application of new generalized probability density function and its applications

The exponential‐based probability density functions (PDFs) such as Gaussian, exponential, and Rayleigh are widely used in lifetime modeling, survival analysis, and engineering. Recently, the generalized probability density functions consisting a well‐known exponential PDFs were introduced and successfully applied in electrical and electronic engineering. After encouraging results of generalized PDF, further theories and applications are considered within this paper. The parameters of the new (GE) PDF were successfully derived by using the maximum likelihood estimation method. The applications in the paper show that the new generalized probability density function based on maximum likelihood parameter estimators provides more effective probabilistic representation. In this study, statistical properties of new GE distribution is discussed. Some reliability characteristics such as hazard function, mean residual lifetime (MRL) function, and mean past lifetime (MPL) functions are obtained based on special functions. Also, r$r$th moment of the new GE random variable is obtained in analytical form. Maximum likelihood estimates (MLEs) of parameters are also discussed. Lastly, some illustrative examples in engineering and survival analysis are provided based on different real data sets.