Entropy measures the amount of uncertainty and dispersion of an unknown or random quantity, this concept introduced at first by Shannon (1948), it is important for studies in many areas. Like, information theory: entropy measures the amount of information in each message received, physics: entropy is the basic concept that measures the disorder of the thermodynamical system, and others. Then, in this paper, we introduce an alternative measure of entropy, called 𝐻𝑁- entropy, unlike Shannon entropy, this proposed measure of order α and β is more flexible than Shannon. Then, the cumulative residual 𝐻𝑁- entropy, cumulative 𝐻𝑁- entropy, and weighted version have been introduced. Finally, comparison between Shannon entropy and 𝐻𝑁- entropy and numerical results have been introduced.

The objective of this study was to present a novel bivariate distribution, which we denoted as the bivariate odd generalized exponential gompertz(BOGE-G) distribution. Other well-known models included in this one include the gompertz, generalized exponential, odd generalized exponential, and odd generalized exponential gompertz distribution. The model introduced here is of Marshall-Olkin type [16]. The marginals of the new bivariate distribution have odd generalized exponential gompertz distribution which proposed by [7]. Closed forms exist for both the joint probability density function and the joint cumulative distribution function. The bivariate moment generating function, marginal moment generating function, conditional distribution, joint reliability function, marginal hazard rate function, joint mean waiting time, and joint reversed hazard rate function are some of the properties of this distribution that have been discussed. The maximum likelihood approach is used to estimate the model parameters. To demonstrate empirically the significance and adaptability of the new model in fitting and evaluating real lifespan data, two sets of real data are studied using the new bivariate distribution. Using the software Mathcad, a simulation research was conducted to evaluate the bias and mean square error (MSE)characteristics of MLE. We found that the bias and MSE decrease as the sample size increases.