Extended Rayleigh Probability Distribution to Higher Dimensions

In this paper, we have derived and studied new probability distributions by extending the 2-dimensional Rayleigh distribution (RD). First, we extend the RD to 3 dimensions and then generalize it to k dimensions for any positive integer k ≥ 3. The distributions are named the 3-dimensional Rayleigh distribution (3-DRD) and k-dimensional Rayleigh distribution (k-DRD), respectively. For both 3-DRD and k-DRD, detailed mathematical and statistical properties including derivations of the corresponding cumulative distribution, probability density, survival, and hazard functions, moments, moment generating functions, mode, skewness, kurtosis, and differential entropy are obtained in closed forms. Parameter estimation is done for both models using the maximum likelihood estimation method and some statistical properties of the estimator are discussed for each case. Interestingly, the commonly known Normal, Rayleigh, Maxwell–Boltzmann, chi-square, gamma, and Erlang distributions are related to the newly developed extended RDs as special cases. For the 3-DRD, plots of cumulative distribution, probability density, survival, and hazard functions are exhibited, a simulation study is carried out, and random samples are generated using the standard accept–reject (AR) algorithm to check the efficiency of the maximum likelihood estimates of the parameter. Moreover, the new 3-DRD model is fitted to one simulated and three real datasets, revealing good performance compared to four existing Rayleigh-based distributions. This study will contribute new knowledge to the field of applied statistics and probability, and the findings will be used as a basis for future research in the field.