Variables random

A random variable is a real-valued frmction defined over die sample space S of a random e, periment (Note diat diis application of probability dieorem to plant and equipment failures, i.e., accidents, requires diat die failure occurs randomly. 

Let X denote die number of die tlirow on wliich die first failure of a switch occurs. Then X is a discrete random variable widi range 1, 2, 3,. .., n,. ... Note duit die nuige of X consists of a countable infinitude of values and diat X is dierefore discrete. 

Suppose that X denotes the time to failure of a bus section in an electrostatic precipitator. Then X is a continuous random variable whose range consists of the real numbers greater dian zero. 

Defining a random variable on a sample space S amounts to coding tlie outcomes in real numbers. Consider, for example, the random experiment involving die selection of an item at randoni from a manufactured lot. Associate X = 0 widi die drawing of a non-defective item and X = 1 widi die drawing of a defective item. Tlien X is a randoni variable with range (0, 1) and dierefore discrete. 

This result can be used to update the prior probabilities of mutually exclusive events B, in light of the new information that A has occurred. The following example 

Example 2.6 Bayes s Rule. A novel biomarker (or a combination of biomarkers) diagnosis assay is 95% effective in detecting a certain disease when it is present. The test also yields 1% false-positive result. If 0.5% of the population has the disease, what is the probability a person with a positive test result actually has the disease  

the probability that a person really has the disease is only 32.3%  

In Example 2.6, how can we improve the odds of detecting real positives We can improve by using multiple independent diagnosis assays, for example. A, B, and C, all with 32.3% detection probabilities. Then, the probability that a person with positive results from all three assays has the disease will be 

A random variable (abbreviated as r.v.) associates a unique numerical value with each outcome in the sample space. Formally, a r.v. is a real-valued function from a sample space S into the real numbers. We denote a r.v. by an uppercase letter (e.g., Xor T) and a particular value taken by a r.v. by the corresponding lowercase letter (e.g., x or y). 

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If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Analysis of tlie global statistics of protein sequences has recently allowed light to be shed on anotlier puzzle, tliat of tlie origin of extant sequences [170]. One proposition is tliat proteins evolved from random amino acid chains, which predict tliat tlieir length distribution is a combination of the exponentially distributed random variable giving tlie intervals between start and stop codons, and tlie probability tliat a given sequence can fold up to fonii a compact... 

We can pick the initial random variables for the classical coordinates and momenta in the way it is done in an ordinary classical trajectory program. 

Fig. 10.19 The probability density of the extreme value distribution typical of the MSP scores for random sequena The probability that a random variable with this distribution has a score of at least x is given by 1 - exp[-e -where u is the characteristic value and A is the decay constant. The figure shows the probability density function (which corresponds to the function s first derivative) for u = 0 and A = 1.

Understanding the distribution allows us to calculate the expected values of random variables that are normally and independently distributed. In least squares multiple regression, or in calibration work in general, there is a basic assumption that the error in the response variable is random and normally distributed, with a variance that follows a ) distribution. 

Example 3 illustrated the use of the normal distribution as a model for time-to-failure. The normal distribution has an increasing ha2ard function which means that the product is experiencing wearout. In applying the normal to a specific situation, the fact must be considered that this model allows values of the random variable that are less than 2ero whereas obviously a life less than 2ero is not possible. This problem does not arise from a practical standpoint as long a.s fija > 4.0. 

Basic Statistical Properties. The PDF for an exponentially distributed random variable t is given by... 

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). 

Random Variables Applied statistics deals with quantitative data. In tossing a fair coin the successive outcomes woula tend to be... 

Counts and measurements are charac terized as random variables, that is, observations which are susceptible to chance. Virtually all quantitative data are susceptible to chance in one way or another. 

Models Part of the foundation of statistics consists of the mathematical models which characterize an experiment. The models themselves are mathematical ways of describing the probabihty, or relative likelihood, of observing specified values of random variables. For example, in tossing a coin once, a random variable x could be defined by assigning to x the value I for a head and 0 for a tail. Given a fair coin, the probabihty of obsei ving a head on a toss would be a. 5, and similarly for a tail. Therefore, the mathematical model governing this experiment can be written as... 

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. 

Characterization of Chance Occurrences To deal with a broad area of statistical apphcations, it is necessary to charac terize the way in which random variables will varv by chance alone. The basic-foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. 

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. 

Nature In some types of engineering and management-science problems, we may be concerned with a random variable which represents a proportion, for example, the proportional number of defective items per day. The method described previously relates to a single proportion. In this subsection two proportions will be considered. 

Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a hnear combination of the measured variables [yj], Y = CL y + Cioyo + Let the random variables yi, yo,. . . have means E yi), E y, . . . and variances G yi), G y, . The variable Y has mean... 

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... 

Typieally, if the stress or strength has not been taken direetly from the measured distribution, it is likely to be a eombination of random variables. For example, a... 

We need this speeial algebra to operate on the engineering equations as part of probabilistie design, for example the bending stress equation, beeause the parameters are random variables of a distributional nature rather than unique values. When these random variables are mathematieally manipulated, the result of the operation is another random variable. The algebra has been almost entirely developed with the applieation of the Normal distribution, beeause numerous funetions of random variables are normally distributed or are approximately normally distributed in engineering (Haugen, 1980). 

Table 4.4 Mean and standard deviation of statistieally independent and eorrelated random variables v and y for some eommon funetions...

The varianee equation ean be solved direetly by using the Calculus of Partial Derivatives, or for more eomplex eases, using the Finite Difference Method. Another valuable method for solving the varianee equation is Monte Carlo Simulation. However, rather than solve the varianee equation direetly, it allows us to simulate the output of the varianee for a given funetion of many random variables. Appendix XI explains in detail eaeh of the methods to solve the varianee equation and provides worked examples. 

The varianee for any set of data ean be ealeulated without referenee to the prior distribution as diseussed in Appendix I. It follows that the varianee equation is also independent of a prior distribution. Here it is assumed that in all the eases the output funetion is adequately represented by the Normal distribution when the random variables involved are all represented by the Normal distribution. The assumption that the output funetion is robustly Normal in all eases does not strietly apply, partieularly when variables are in eertain eombination or when the Lognormal distribution is used. See Haugen (1980), Shigley and Misehke (1996) and Siddal (1983) for guidanee on using the varianee equation. 

Theoretically, the effects of the manufacturing process on the material property distribution can be determined, shown here for the case when Normal distribution applies. For an additive case of a residual stress, it follows that from the algebra of random variables (Carter, 1997) ... 

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