Sampling random variable

Precision is a term which must be handled with care because there are many different precisions. Any time there is a source of variability, there is a precision associated with it. It is usually expressed as a standard deviation at a certain level of analyte. It can be associated with sampling as a random variability within a single material or as an among samples random variability of a number of related materials. The most common analytical precision terms are repeatability, which is the term associated with a single operator (within-laboratory) and reproducibility, which is the term associated with different operators in different laboratories (between-laboratory). For research work, repeatability is most often reported for regulatory work, the variability between laboratories is the most important. The... 

Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as X, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like [L and O are not random variables they are fixed constants. 

Determining the area under the normal cuiwe is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x — l)/<7, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity = (x-[l)/ G/Vn). 

Nature In some types of applications, associated pairs of obseiwa-tions are defined. For example, (1) pairs of samples from two populations are treated in the same way, or (2) two types of measurements are made on the same unit. For applications or tnis type, it is not only more effective but necessary to define the random variable as the difference between the pairs of observations. The difference numbers can then be tested by the standard t distribution. 

Applieation for approximately 40 years Design parameters treated as random variables Small samples used to obtain statistieal distributions... 

In reality, it is impossible to know the exaet eumulative failure distribution of the random variable, beeause we are taking only relatively small samples of the... 

Monte Carlo simulation is a numerical experimentation technique to obtain the statistics of the output variables of a function, given the statistics of the input variables. In each experiment or trial, the values of the input random variables are sampled based on their distributions, and the output variables are calculated using the computational model. The generation of a set of random numbers is central to the technique, which can then be used to generate a random variable from a given distribution. The simulation can only be performed using computers due to the large number of trials required. 

A random variable is a real-valued function defined over tlie sample space S of a random experiment (Note tliat tliis application of probability tlieorem to plant and equipment failures, i.e., accidents, requires tliat tlie failure occurs randomly. 

Tlie mean )i and tlie variance a" of a random variable are constants cliaracterizing die random variable s average value and dispersion about its mean. The mean and variance can be derived from die pdf of the random variable. If die pdf is miknown, however, the mean and die variance can be estimated on die basis of a random sample of observations on die random variable. Let X, Xj,. X, denote a random sample of n observations on X. 

In tlie case of a random sample of observations on a continuous random variable assumed to have a so-called nonnal pdf, tlie graph of which is a bellshaped curve, tlie following statements give a more precise interpretation of the sample standard deviation S as a measure of spread or dispersion. 

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

Tlie probabilities given in Eqs. (20.5.10), (20.5.11), and (20.5.12) are tlie source of the percentages cited in statements 1, 2, and 3 at tlie end of Section 19.10. These can be used to interpret tlie standard deviation S of a sample of observations on a normal random variable, as a measure of dispersion about tlie... 

Tlie nonnal distribution is used to obtain probabilities concerning tlie mean X of a sample of n observations on a random variable X. If X is nonnally distributed witli mean p and standard deviation a, tlien X, tlie sample mean, is nonnally distributed witli mean p. and standard deviation. For example, suppose X is nonnally distributed witli mean 100 and standard deviation 2. 

Estimates of the parameters a and p in tlie pdf of a random variable X having a log-normal distribution can be obtained from a sample of observations on X by making use of tlie fact diat In X is normally distributed with mean a and standard deviation p. Tlierefore, tlie mean and standard deviation of the natural logaritluns of tlie sample observations on X furnish estimates of a and p. To illustrate tlie procedure, suppose the time to failure T, in thousands of hours, was observed for a sample of 5 electric motors. The observed values of T were 8, 11, 16, 22, and 34. The natural logs of these observations are 2.08, 2.40, 2.77, 3.09, and 3.53. Assuming tliat T has a log-normal distribution, the estimates of the parameters a and p in the pdf are obtained from the mean and standard deviation of the natural logs of tlie observations on T. Applying the Eqs. (19.10.1), and (19.10.2) yields 2.77 as tlie estimate of a and 0.57 as tlie estimate ofp. 

The moments describe the characteristics of a sample or distribution function. The mean, which locates the average value on the measurement axis, is the first moment of values measured about the origin. The mean is denoted by p for the population and X for the sample and is given for a continuous random variable by... 

The skew, the third moment about the mean, is a measure of symmetry of distribution and can be denoted by y (population) or g (sample). It is given for a continuous random variable by... 

Mutual information is thus a random variable since it is a real valued function defined on the points of an ensemble. Consequently, it has an average, variance, distribution function, and moment generating function. It is important to note that mutual information has been defined only on product ensembles, and only as a function of two events, x and y, which are sample points in the two ensembles of which the product ensemble is formed. Mutual information is sometimes defined as a function of any two events in an ensemble, but in this case it is not a random variable. It should also be noted that the mutual... 

A Markov chain is a sequence of trials that samples a random variable and satisfies two conditions, namely that the outcome of each trial belongs to a finite set of outcomes nd that the outcome of each trial depends only on the... 

The probability of a given event is often represented as a function of a random variable, say, x. The random variable can take on various discrete values Xj with probabilities given by W x-,). The variable jt is then an independent variable that describes a random or stochastic process. The function W(Xf) in simple examples is discontinuous, although as the number of samples increase, it approaches a denumerable infinity. 

The principle of sequential analysis consists of the fact that, when comparing two different populations A and B with pre-set probabilities of risks of error, a and /3, just as many items (individual samples) are examined as necessary for decision making. Thus the sample size n itself becomes a random variable. 

Correlation analysis investigates stochastic relationships between random variables on the basis of samples. The interdependence of two variables x... 

Random Variables Applied statistics deals with quantitative data. In tossing a fair coin the successive outcomes would tend to be different, with heads and tails occurring randomly over a period of time. Given a long strand of synthetic fiber, the tensile strength of successive samples would tend to vary significantly from sample to sample. 

Most techniques for process data reconciliation start with the assumptions that the measurement errors are random variables obeying a known statistical distribution and that the covariance matrix of measurement errors ( J>) is given. In contrast, in this chapter we discuss direct and indirect approaches for estimating the variances of measurement errors from sample data. Furthermore, a robust strategy is presented for dealing with the presence of outliers in the data set. 

Let x, x2,..., xn be a random sample of N observations on a random variable x with exponential density function... 

The estimation of means, variances, and covariances of random variables from the sample data is called point estimation, because one value for each parameter is obtained. By contrast, interval estimation establishes confidence intervals from sampling. 

In constructing confidence intervals, it is essential to use suitable random variables whose values are determined by the sample data as well as by the parameters, but whose distributions do not involve the parameters in question. 

Collection of a sample of size N from the random variable X. 

In words, the right-hand side is the probability that the random variable U (x, t) falls between the sample space values V) and V) + dV) for different realizations of the turbulent flow.5 In a homogeneous flow, this probability is independent of x, and thus we can write the one-point PDF as only a function of the sample space variable and time /(Vi i ). 

Alternatively, an LES joint velocity, composition PDF can be defined where both (j> andU are random variables Aj 0 U 4 U >4 x, t). In either case, the sample space fields U and0 are assumed to be known. 

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. 

The set of all possible outcomes of a measurement considered as a random variable is usually called the population. The parameters of the density function associated with a particular population, e.g., mean or variance, are not physically accessible since their determination would require an infinite number of measurements. A measurement, or more commonly a set of measurements ( points or observations ), produces a finite set of outcomes called a sample. Any convenient number describing in a compact form some property of the sample is called a statistic, e.g., the sample mean... 

---