A single random variable

In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite reservoir or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964 Hoel et al., 1971 Lloyd, 1980 Papoulis, 1984 Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to the field of continuous random variables. 

Because F(x) is non-decreasing, its derivative /(x) is non-negative. Conversely, if fi is the domain ] — oo, + oo[, the cumulative distribution function F(x) relates to the probability density function /(x) through 

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Parallel to the case of a single random variable, the mean vector and covariance matrix of random variables involved in a measurement are usually unknown, suggesting the use of their sampling distributions instead. Let us assume that x is a vector of n normally distributed variables with mean n-column vector ft and covariance matrix L. A sample of m observations has a mean vector x and annxn covariance matrix S. The properties of the t-distribution are extended to n variables by stating that the scalar m(x—p)TS ( —p) is distributed as the Hotelling s-T2 distribution. The matrix S/m is simply the covariance matrix of the estimate x. There is no need to tabulate the T2 distribution since the statistic... 

The same note of caution applies to strongly non-linear functions j as in the case of a single random variable. 

In many experiments there will be more than a single random variable of interest, say Xj, X2, X3,. .. etc. These variables can be conceptualized as a k-dimen-sional vector that can assume values (xj, X2, X3,... etc). For example, age, height, weight, sex, and drug clearance may be measured for each subject in a study or drug concentrations may be measured on many different occasions in the same subject. Joint distributions arise when there are two or more random variables on the same probability space. Like the one-dimensional case, a joint pdf is valid if... 

For many problems there are situations involving more than a single random variable. Consider the case of 2 discrete random variables, x and y. Here y may take any of the values of the set +". Now instead of having PMFs of a single variable, the joint PMF,P(x,y) is required. This may be viewed as the probability of x taking a particular values and y taking a particular value. This joint PMF must satisfy... 

For a single random variable the maximum entropy distribution is obtained by considering only the moment constraints. For the multivariate distribution the correlations between each pair of random variables has to be taken into account as well. This would lead to an optimization problem with 4 optimization parameters with 4 constraints from the marginal moment conditions and ( 4-1)/2 constraints from the correlation conditions, where n is number of random variables. This concept was recently apphed in Soize (2008) to determine the joint density function. For a... 

It is quite common to describe the demand with random variables and fiizzy numbers, but in the complicated market surroundings, it is not accurate enough to describe the demand with a single random variable or fiizzy variable under many circumstances. For example, the future market demand might be good, or moderate, or bad, these three cases have happened in a certain probabihty, that is to say it is stochastic, but the market demand after each possibility is a fuzzy process, for example, when the market is good, the demand might be around 5000 units when the market is moderate, it is around 2600 units. In the bad case, it is only around 900 units. These descriptions are fuzzy, which means the customer demand is a fuzzy random variable. 

Autocorrelation Coefficient n The auto covariance normalized by the product of the standard deviations of the two sections from the single random variable sequence used to calculate the autocovariance. In other words the autocorrelation coefficient is the cross-correlation coefficient of two sub-sequences of the same random variable. It is probably the most commonly used measure of the correlation between two sections of a single random variable sequence. It is often simply but incorrectly referred to as the autocorrelation, which is the un-normalized expectation value of the product of the two sequence sections. The autocorrelation coefficient of the two subsequences of random variable, X, is often denoted by Pxx(ii>T) where i is the starting index of the second section, and T is the length of the sections. The precise mathematical definition of autocorrelation coefficient of two random variable sequence sections is given by ... 

The autocorrelation is the cross-correlation of a subsequence of a single random variable with another suh-sequence of the same random variable. 

The mathematical model for a discrete stochastic process is a sequence of random variables for n = 0,1,2,..., where is the state at time n. Here the superscript (n) indicates time point n. All knowledge about a single random variable is in its probability distribution. Similarly, all knowledge about the probabilistic law for a stochastic process is contained in the joint probability distribution of every... 

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