The Distribution of Frequently Used Random Variables

The distribution of a population s property can be introduced mathematically by the repartition function of a random variable. It is well known that the repartition function of a random variable X gives the probability of a property or event when it is smaller than or equal to the current value x. Indeed, the function that characterizes the density of probability of a random variable (X) gives current values between X and x -I- dx. This function is, in fact, the derivative of the repartition function (as indirectly shown here above by relation (5.16)). It is important to make sure that, for the characterization of a continuous random variable, the distribution function meets all the requirements. Among the numerous existing distribution functions, the normal distribution (N), the chi distribution (y ), the Student distribution (t) and the Fischer distribution are the most frequently used for statistical calculations. These different functions will be explained in the paragraphs below. 

The famous normal distribution can be described with the following example a chemist carries out the daily analysis of a compound concentration. The samples studied are extracted from a unique process and the analyses are made with identical analytical procedures. Our chemist observes that some of the results are 

The graphic construction of this computation is given in Fig. 5.4. Two examples are given the first concerns the processing of 50 samples and the second the processing of 100 samples. When the mean value of the processed measurements has been computed, we can observe that it corresponds to the measurement that has the maximum value of apparition frequency. The differences observed between the two measurements are the consequence of experimental errors [5.16, 5.17]. Therefore, all the measurement errors have a normal distribution written as a density function by the following relation  

Here p and are, respectively, the mean value and the dispersion (variance) with respect to a population. These characteristics establish all the integral properties of the normal random variable that is represented in our example by the value expected for the species concentration in identical samples. It is not feasible to calculate the exact values of p and because it is impossible to analyse the population of an infinite volume according to a single property. It is important to say that p and show physical dimensions, which are determined by the physical dimension of the random variable associated to the population. The dimension of a normal distribution is frequently transposed to a dimensionless state by using a new random variable. In this case, the current value is given by relation (5.21). Relations (5.22) and (5.23) represent the distribution and repartition of this dimensionless random variable. Relation (5.22) shows that this new variable takes the numerical value of x when the mean value and the dispersion are, respectively, p = 0 and = 1. 

Before presenting some properties of normal distribution, we have to present the relation (5.25) that gives the probability for which one random variable is fixed between a and b values (a b), with the repartition function  

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