‎In this article‎, ‎the method of determining the optimal sample size is based on Linex asymmetric loss function and has been expressed through Bayesian method for normal‎, ‎Poisson and exponential distributions‎. ‎The desirable sample size has been calculated through numerical method‎. ‎In numerical method‎, ‎the average posterior risk is calculated and then it is added to the Lindley linear cost function to achieve the average of the total cost‎. ‎Then‎, ‎the diagram of sample size is drawn in comparison to the average of total cost and eventually‎, ‎the optimal sample size that minimizes the cost has been achieved.

‎In this paper‎, ‎a new distribution of the three-parameter lifetime model called the Marshall-Olkin Gompertz is proposed on the basis of the Gompertz distribution‎. ‎It is a generalization of the Gompertz distribution having decreasing failure rate and can also be increasing and bathtub-shaped depending on its parameters‎. ‎The probability density function‎, ‎cumulative distribution function‎, ‎hazard rate function and some mathematical properties of this model such as‎, ‎central moments‎, ‎moments of order statistics‎, ‎Renyi and Shannon entropies and quantile function are derived‎. ‎In addition‎, ‎the maximum likelihood of its parameters method is estimated and this new distribution compared with some Gompertz distribution generalizations by means of a set of real data‎. 

‎In this paper we propose a utility function and obtain the Bayese stimate and the optimum sample size under this utility function‎. ‎This utility function is designed especially to obtain the Bayes estimate when the posterior follows a gamma distribution‎. ‎We consider a Normal with known mean‎, ‎a Pareto‎, ‎an Exponential and a Poisson distribution for an optimum sample size under the proposed utility function‎, ‎so that minimizes the cost of sampling‎. ‎In this process‎, ‎we use Lindley cost function in order to minimize the cost‎. ‎Here‎, ‎because of the complicated form of computation‎, ‎we are unable to solve it analytically and use the mumerical methids to get the optimum sample size.