‎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.

One of the challenges for decision-makers in insurance and finance is choosing the appropriate criteria for making decisions. Mathematical expectation, expected utility, and distorted expectation are the three most common measures in this area. In this article, we study these three criteria, and by providing some examples, we review and compare the decisions made by each measure.