## 1 Introduction  
In many areas of human presence, situations arise when a subject is interested in comparing different scenarios they can choose from, the scenarios being related to their activity. The objective is to opt for the right scenario to ensure that a proper strategy is pursued in the future. The output of each scenario is often expressed by a measure that can take on only positive values. The measure allows one to reflect on the consequences of the scenarios so that a proper decision can ultimately be made in advance. However, making that right decision is often not a straightforward task despite the introduced measure because each scenario manifests itself in the end with more than one value of the measure due to its dependence on random events, and so a more rigorous, statistical approach should be adopted prior to making the selection. Let us look at some important real-life examples of such situations. In industries, a quality characteristic of a product can be monitored, for instance, and two different production technologies might be capable of fabricating the product. The company's management wants to know which technology leads to a more reliable output as far as the stability of the observed characteristic is concerned. To tackle this problem, a variability of the characteristic can be measured for each production batch to assess the reliability or stability in question. The variability will generally be different for each batch and can therefore be regarded as a random variable, variable X for production technology 1 and variable Y for production technology 2. The question then is whether the two production reliabilities are the same in the sense that their expected values are the same: \(E(X)=E(Y)\). If this is true, the company's management can go for the cheaper of the two technologies since the more expensive one will not bring anything new regarding reliability. The ultimate result of this decision will then be financial savings. In finance, an insurance company may strive to decide which of two specific and different insurance products it should offer to its customers. It will naturally prefer the one that exposes it to lesser risk, the risk being measured by how much money it may need to pay out within its insurance coverage liabilities. Here, the amount paid out due to an event covered by an insurance product represents a positive random variable, the value of which differs from one event to another, and so the company can ask itself the question, \"Do we spend, on average, the same amount of money on the two insurances?\"  
An investor cares about the yield of a financial investment of a certain type; the yield is always observed after a specified period of time. The yield history has so far been positive values for the investments. Such an investor may look at the past yields to decide which investment to go for by answering the question \"Do the various investments provide the same average yield for the observed time period?\"  
In seismology, the strength of tremor is certainly one of many positive variables of interest, and it can be accompanied by the question: \"Do the two analyzed geographic regions experience earthquakes of the same magnitude, on average?\"  
The examples above work with two positive random variables \(X\) and \(Y\), and it could also be said without assuming too much that one of their distinctive features is that they take on lower values probably more often than very high values. This is true for all the given examples. Thus, a right-skewed probability distribution, such as the gamma distribution [1], might be an appropriate probabilistic model for the description of the variables. If such a model is acceptable, the uttered questions come down to comparing the expected values of two gamma-distributed random variables. The standard procedure that usually follows in these cases takes the form of an assessment of whether the expression \(\overline{X}-\overline{Y}\), or a transformation thereof, is too far from a set constant, usually zero.