Independent and identically distributed

Let SN( = X + + XNt be a sum of random variables with random subscript Nt independently and identically distributed (according to F(x)). Thus N is also a random variable, and it depends on t. We wish to compute the expected value and the variance of SNt. We have... 

The amount accepted for stocking is the minimum of R and the quantity to satisfy back-logged demand bringing the level up to 8. The demand in the i 1 period is given by a random variable , with continuous density function ( ) , all variables ,(t = 1,2, ) are independently and identically distributed. The level of stock at the end of period i is represented by the random variable Xt measured before adding any delivery occurring at time i. Let the random variable be the time of the t1 delivery. Then Prob (17, = 0) = 1. We can write X t and Rm to indicate dependence on ij. We have ... 

Consider the following inventory problem. There are p time periods at the start of each of which an order of n items is made at a cost A(n), which is an increasing function of % (e.g., A(n) = a + bn). The length of each period is a random variable, and, hence, there are p random variables Xt (i = 1,- -, p) that are assumed to be independently and identically distributed according to the distribution function Fn(x)—for each period, it is the probability that there is a demand for... 

Even if we make the stringent assumption that errors in the measurement of each variable ( >,. , M.2,...,N, j=l,2,...,R) are independently and identically distributed (i.i.d.) normally with zero mean and constant variance, it is rather difficult to establish the exact distribution of the error term e, in Equation 2.35. This is particularly true when the expression is highly nonlinear. For example, this situation arises in the estimation of parameters for nonlinear thermodynamic models and in the treatment of potentiometric titration data (Sutton and MacGregor. 1977 Sachs. 1976 Englezos et al., 1990a, 1990b). 

It is important to keep in mind that demands occur on the time line. If a reasonable minimal time interval is chosen, it is in many cases justified to consider the demands in these intervals as independent and identically distributed random variables. This means for example that the demand per week is the iterated convolution of the daily demand. A large customer base is a good indicator of independent random demand in different time intervals. 

For any R independent and identically distributed sample batches (denoting the number of sample replication) each with sample size of N, that is,. .., %N,j = 1,..., R, the... 

The KS limits make no distributional assumptions, but they do require that the samples are independent and identically distributed. Additional distributional assumptions can be made that could tighten the KS limits. For instance, assuming the nnderlying distribntion from which the samples came is normal yields a much tighter p-box. In practice, the assnmption abont independence of the individual samples may sometimes be hard to justify, such as when contamination hotspots are the focns of targeted sampling efforts. Techniqnes to account for nonrandom sampling are a topic of cnrrent research. 

Suppose we change the assumptions of the model in Section 5.3 to AS5 (x ) are an independent and identically distributed sequence of random vectors such that x, has a finite mean vector, finite positive definite covariance matrix Zxx and finite fourth moments E[xjxj xixm] = for all variables. How does the proof of consistency and asymptotic normality of b change Are these assumptions weaker or stronger than the ones made in Section 5.2 ... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

We assume a fixed total population N = 217 to which a GCKS provides key management, i.e., the number of potential members is 217. Each member independently decides to join or leave the group. We approximate the members arrival by a fixed-rate Poisson process and assume that the lifetimes are independently and identically distributed random variables. In the simulation, we evaluate different lifetime distribution... 

Second, assume we estimate using an AR(1) process with rj( = pr)t, + y with y independent and identical distributed. 

The location-and-scale model states that the n univariate observations x are independent and identically distributed (i.i.d.) with distribution function F[Qr - 6)/o], where F is known. Typically F is the standard Gaussian distribution function O. 

The random hazard rate model is easily obtained from the above by considering a single unit, mo = 1, and no particles initially administered into the system. The first two moments are obtained by summing n0 independent and identically distributed experiments ... 

Boltzmann distribution — The Boltzmann distribution describes the number N, of indistinguishable particles that have energy , after N of them have been independently and identically distributed among a set of states i. The probability density function is... 

Expanding to the lowest-order nonvanishing term, averaging the result obtained, and assuming that the sample points are independent and identically distributed, we obtain... 

The Central-Limit Theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend toward the normal distribution, equation (3.17), as the sample size becomes large. ... 

Most metamodel techniques assume that the errors (e) are independent and identically distributed (with a normal distribution) f W(0,C7 ) Vx. However, the errors in the predicted values are usually not independent, but they are a function of x. Therefore, the kriging fitting approach is comprised of two parts a polynomial term and a departure from that polynomial ... 

The original data are assumed to be an independent and identically distributed sample of size m, from an unknown probability distribution, G (xi,... 

Some statistics concepts such as mean, range, and variance, test of hypothesis, and Type I and Type II errors are introduced in Section 2.1. Various univariate SPM techniques are presented in Section 2.2. The critical assumptions in these techniques include independence and identical distribution [iid) of data. The independence assumption is violated if data are autocorrelated. Section 2.3 illustrates the pitfalls of using such SPM techniques with strongly autocorrelated data and outlines SPM techniques for autocorrelated data. Section 2.4 presents the shortcomings of using univariate SPM techniques for multivariate data. 

The distribution function P x,) of a given line amplitude is itself two-dimensional because the amplitude x, is, in general, complex. Since only the modulus of xf is constrained by the data on one-photon absorption, the real and imaginary parts of x, are independently and identically distributed. 

It can be shown (see, for example, Cramer, 1991, pp. 50-1) that given an assumption that the random disturbances in equation (3.1) are independently and identically distributed (i.i.d.) according to a Weibull distribution , the probability that theyth treatment is selected can be written as ... 

Binomial Distribution Some experiments can be viewed as a sequence of independent and identically distributed (i.i.d.) Bernoulli trials, where each outcome is a success or a Tailure. The total number of successes from such an experiment is... 

Model 1 Zi,..., are independent and identically distributed observations from a Bernoulli distribution with probability p. These Bernoulli random variables take the value 0 and 1. 

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