Variables random, defined

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

No simple form of the moment generating function exists. In the special case where 0C =a2 = 1, the beta distribution reduces to the uniform distribution over [0, 13- Finally, we will frequently refer to Snedecor s F-distribution. A random variable defined over ]0, + 00 [ is distributed with the F-distribution with v, and v2 degrees of freedom... 

This is a random variable defined over the space of the two conformational states L and H. 

The random variables defined above, except for SK, are always finite, because the running time of the corresponding algorithms is deterministically polynomial. Moreover, any lower bound on the length of the values of is also valid for SK. 

The random variable defined by = (n — follows a distribution, also... 

The Z-distribution and Eq. 13 are applicable w hen both the population mean and the variance are known. When the variance must be estimated from a sample, the r -distribution and the proportion of sampling values that fall within the — and -f- limits as given above no longer apply. In such circumstances a distribution called Student s /-distribution is used, and / is a random variable defined by Eq. 17. 

Mills [107] concentrates on showing the effect of uncertainty on a monopolist s short-run pricing policy under the assumption of additive demand. In his model, the demand is specified as D p, s) = y p) -f- e,where y p) is a decreasing function of price p and e is a random variable defined within some range. In particular, he shows that the optimal price under stochastic demand is always no greater than the optimal price under the assumption of deterministic demand, called the riskless price. Lau and Lau [90] and Polatoglu [122] both study different cases of demand process for linear demand case where y p) — a bp, where a, 6 > 0. 

Interpretation. Let Tj, T2,..T . be a sequence of independent random variables with y(l, v) exponential distribution. Then the random variable defined by the sum... 

Density function of a random variable Defined implicitly the probability that the random variable lies between a and b is the area under the density function between a and b. This approach introduces the methods of calculus and analysis into probability. 

In order to show the potential use of the methodology, the distribution function of two random variables defined in terms of the final shoreline position is calculated next for p = 2. 

Example 5.2. Consider the reliability problem from (Choi, Grandhi, Canfield 2010), with X = X, X2), where X, jV(10,5) and Xj—.AfClO.S) are independent random variables defined in 2 = R, and limit-state function g(xj,X2) = Xj H-2x2 — 20. The problem is formulated as a PC problem space ... 

Example 5.3. Consider the reliabihty assessment problems from (Sorensen 2004, Note 6) (series system) and (Sorensen 2004, Note 7) (parallel system), with X = (X,X2), where Xj and Xj are independent standard normal random variables defined in 2 = R, and the limit state functions of Table 3. 

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. 

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. 

In estimating the fragility parameters, it is convenient to work in terms of an intermediate random variable known as the factor of safety F. This is defined as the ratio of the grouiiti-aeederation capacity A to the safe shutdown earthquake (SSE) acceleration used in plant tlesign. 

After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

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. 

Anotlier fimction used to describe tlie probability distribution of a random variable X is tlie cumulative distribution function (cdf). If f(x) specifies tlie pdf of a random variable X, tlien F(x) is used to specify the cdf For both discrete and continuous random variables, tlie cdf of X is defined by ... 

Note that tlie cdf is defined for all real numbers, not just tlie values assmiied by tlie random variable. 

Monte Carlo simulation uses computer programs called random number generators. A random number may be defined as a nmnber selected from tlie interval (0, 1) in such a way tliat tlie probabilities that the number comes from any two subintervals of equal lengtli are equal. For example, the probability tliat tlie number is in tlie subinter al (0.1, 0.3) is the same as the probability tliat tlie nmnber is in tlie subinterval (0.5, 0.7). Random numbers thus defined are observations on a random variable X having a uniform distribution on tlie interval (0, 1). Tliis means tliat tlie pdf of X is specified by... 

We begin our discussion of random processes with a study of the simplest kind of distribution function. The first-order distribution function Fx of the time function X(t) is the real-valued function of a real-variable defined by6... 

The characteristic function of the random variable (or, equivalently, of the distribution function F ) is defined to be the expectation of the random variable eiv4> where v is a real number.16 In symbols... 

Complex valued random variables will be discussed in Section 3.8. For the time being it is sufficient to state that the expectation of a complex function is defined by E[fa = E[fa] + iE[fa] where fa and fa denote the real and imaginary parts, respectively, of fa... 

It should be carefully noted that a random variable is only unambiguously defined when the time increments fn and f m are specified... 

The expectation symbol E obeys the same rules of manipulation, Eq. (3-40), as in the one-dimensional case. The only additional comment needed here is that the addition rule holds even when the two random variables concerned are defined with respect to different sets of r s. The proof of this fact is immediate when the various expectations involved are written as time averages. 

We conclude this section by introducing some notation and terminology that are quite useful in discussions involving joint distribution functions. The distribution function F of a random variable associated with time increments fnf m is defined to be the first-order distribution function of the derived time function Z(t) = + fn),... 

A few minutes thought should convince the reader that all our previous results can be couched in the language of families of random variables and their joint distribution functions. Thus, the second-order distribution function FXtx is the same as the joint distribution function of the random variables and 2 defined by... 

Joint Moments and Characteristic Functions.—The joint momenta cckli...ikn of a family of n random variables dm are defined by the expression... 

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