Integrated quadrably reduced additive weighted mixture distributions

We propose new families of probability distributions derived from mixtures of weighted probability distributions. These distributions have not been addressed in the statistics literature. Many new continuous and discrete probability distributions could be generated from these families.We simplify a mixture of probability density (mass) functions into one unique closed probability density (mass) function. Some of the distributions in the families could be used in modeling survival analysis problems in biostatistics and reliability techniques in engineering. These distributions can easily capture different features of research data sets, such as bimodality, symmetry and asymmetry. A new parameter that vertically translates the sum of the weights is introduced. We call this parameter the translation parameter since it controls the number of the modes for the given distribution. Closed form of a generalized exponential distribution that could be modelled on bimodal data is derived from the mixture and employed by fitting real and simulated data sets. The survival and hazard functions of the exponential distribution are also investigated.