Study on the Expectation of Geometric Distribution

When we want to know the truth of the emergence of a scientific principle, we find in many cases that it is no more than a process of interpretation, description and analysis of existing natural phenomena of various kinds, this is done by identifying the components on which these phenomena depend and the qualities and characteristics that each organism enjoys, which enables us to form a clear strategic vision that helps us in developing pre-solutions to what we expect from the problems resulting from any emergency or exceptional circumstance Most studies in the fields of science need two types of study: 1- Study the real data obtained through observation and description, and then record and tabulate the case information we are studying. 2- the study of phenomena to predict what the phenomenon will become in the near and even distant future, The statistical analysis puts us in the merits of the current and future situation of the system under study, and statistical analysis is the applied aspect of probability science, through the probability distributions that are used in different fields of science, the probability of the values of the variable can be calculated by applying a mathematical equation called the probability density function and by using distinctive values for these distributions such as expectation, variance and standard deviation From analyzing the data to reach the desired results, after reviewing a number of references, it drew my attention that the expectation of the geometric distribution is calculated using a relationship that varies from one return to another, knowing that these references start from the basic definition of the expectation or from the moment-generating function, and logically the results must be the same, so I prepared this research, through which I explained the reason for the difference, which in turn may have an impact on the results of studies for the systems which works according to this distribution (the importance of this effect depends on the meaning of the expectation for the studied system)