Normal distribution with known mean

For a normal distribution with known mean, p, and standard deviation, a, the exact proportion of values which lie within any interval can be found from tables, provided that the values are first standardized so as to give z-values. This is done by expressing a value of x in terms of its deviation from the mean in units of standard deviation, a. That is... 

Single Draw from a Normal Distribution with Known Mean... 

Example 3. A centrifugal pump moving a corrosive Hquid is known to have a time-to-failure that is well approximated by a normal distribution with a mean of 1400 h and a standard deviation of 120 h. A particular pump has been in operation for 1080 h. In order to plan maintenance activities the chances of the pump surviving the next 48 h must be deterrnined. 

Therefore, on statistical grounds, if the error terms (e,) are normally distributed with zero mean and with a known covariance matrix, then Q( should be the inverse of this covariance matrix, i.e.,... 

Even with powerful computer programs at hand, the solution of estimation problems is usually far from simple. A convenient way to eliminate computational errors and to study the effects of statistical assumptions is to solve first a problem with known true parameter values, involving data generated at some nominal parameter vector. Initially it is advisable to investigate with error-free data, then to add errors of the assumed structure. The simulation usually requires normally distributed random variables. Random numbers R that approximately are from a normal distribution with zero mean and unit variance can be obtained by... 

Exact analytical solutions can be obtained for summations of normal distributions. The sum of normal distributions is identically a normal distribution. The mean of the sum is the sum of the means of each input distribution. The variance of the sum is the sum of the variance of the inputs. Any statistic of interest for the output can be estimated by knowing its distribution type and its parameters. For example, for a model output that is a normal distribution with known parameter values, one can estimate the 95th percentile of that output. 

Using the example assume that one has data on the source rate (G) which indicates that between residence values are normally distributed with a mean of 50mgh and a standard deviation of 5mgh for the particular source of interest. (This is an example of uncertainty type 1 above - a known and measured quantity with natural variability.)... 

Assume that the random variable X has a normal distribution with an unknown population mean, p, and with a known population variance, For a sample size of n, the sampling distribution of the sample mean has a normal distribution with population mean, p, and variance, The... 

A variety of techniques is nowadays available for the solution of inverse problems [26,27], However, one common approach relies on the minimization of an objective function that generally involves the squared difference between measured and estimated variables, like the least-squares norm, as well as some kind of regularization term. Despite the fact that the minimization of the least-squares norm is indiscriminately used, it only yields maximum likelihood estimates if the following statistical hypotheses are valid the errors in the measured variables are additive, uncorrelated, normally distributed, with zero mean and known constant standard-deviation only the measured variables appearing in the objective function contain errors and there is no prior information regarding the values and uncertainties of the unknown parameters. 

Consider the data shown in Figure 9.13 with two unknown parameters and only 10 data points. The measurement errors are drawn from a normal distributed with zero mean and variance rr = 10 Compute the best estimates of activation energy and mean rate constant and the 95% confidence Intervals for the cases of known. and unknown measurement variance. 

An additional interesting example arises from the normal distribution. Suppose that the demand for a product behaves as a sequence of independent samples from a normal distribution with a mean of /i and standard deviation a. However, even though the parameter a is known, the exact value of /i can only be characterized by a priori normal density N(//q, cto) It can be shown that for a given sequence of demand observations di,..., dt-, a sufficient statistic is 5 = Xli=i that the posterior distribution of the mean demand pi is... 

The formulation of PE problems is usually based on the concept of maximum-likelihood (ML) estimation. Therefore, one obtains objective functions of a rather specific form. Quite often, the observations of the model output y are assumed to be affected with errors that are normally distributed with zero mean and known covariance matrix U. Then, the minimization of the weighted least-squares term where e denotes the difference between y... 

If the measurement is carried out in a way preventing die occurrence of systematic errors, then it is often reasonable to assume that the errors have normal distribution with zero mean and known variance and can be estimated by analysis of the experimental procedure. According to the maximum-likelihood method (77BEC1) the most probable values of the parameters are i... 

Retailers have been known to cancel their orders near the winter season as they have better visibility into expected demand. How many orders should the supplier accept if cancellations are normally distributed, with a mean of 800 and a standard deviation of 400 How many orders should the supplier accept if cancellations are normally distributed, with a mean of 15 percent of the orders accepted and a coefficient of variation of 0.5 ... 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

Parameter Two distinct definitions for parameter are used. In the first usage (preferred), parameter refers to the constants characterizing the probability density function or cumulative distribution function of a random variable. For example, if the random variable W is known to be normally distributed with mean p and standard deviation o, the constants p and o are called parameters. In the second usage, parameter can be a constant or an independent variable in a mathematical equation or model. For example, in the equation Z = X + 2Y, the independent variables (X, Y) and the constant (2) are all parameters. 

To demonstrate the accuracy, two dust and two soil reference materials were analyzed with the described method. The mean value of the correlation coefficients between the certified and the analyzed amounts of the 16 elements in the samples is r = 0.94. By application of factor analysis (see Section 5.4) the square root of the mean value of the communahties of these elements was computed to be approximately 0.84. As frequently happens in the analytical chemistry of dusts several types of distribution occur [KOM-MISSION FUR UMWELTSCHUTZ, 1985] these can change considerably in proportion to the observed sample size. In the example described the major components are distributed normally and most of the trace components are distributed log-normally. The relative ruggedness of multivariate statistical methods against deviations from the normal distribution is known [WEBER, 1986 AHRENS and LAUTER, 1981] and will be tested using this example by application of factor analysis. 

Lets assume that errors e, and 8, are independent values which have normal distributions with mean values equal to zero, and variances o and o, correspondingly. In this case, if ox is known as the standard uncertainty of the measurement standard (CRM), the slope (P,) and the intercept (P0) of the calibration curve can be estimated by the following equations [6] ... 

According to the important theorem known as the central limit theorem, if N samples of size n are obtained from a population with mean, fi, and standard deviation, a, the probability distribution for the means will approach the normal probability distribution as N becomes large even if the underlying distribution is nonnormal. For example, as more samples are selected from a bin of pharmaceutical granules, the distribution of N means, x, will tend toward a normal distribution with mean /j and standard deviation <7- = a/s/n, regardless of the underlying distribution. 

Validation The use of TMAs enables analysis of large data sets, however this does not by any means suggest that the data set is not skewed. This skewing may be the result of the institution s location (population distributions with regards to race, ethnicity, access to health care), type of practice (community hospital versus referral center). These collectively might influence the tumor size, grade and subtype composition of the cases in the dataset. Such abnormalities of the dataset need to be compensated the involvement of a biostatistician from the start (i.e from case selection) helps to prevent the creation of biased TMAs. It is useful to perform common biomarker analysis on sections from the created TMA to confirm the normal distribution of known parameters. Comparison of this data with prior clinical data (e.g. ER analysis) obtained from whole section analysis is particularly useful to validate utility of the TMA. Alternatively the incidence of expression of a number of biomarkers in the TMA should be compared to that in published literature (using whole sections). 

The standard deviation cr of a normally distributed variable y with known mean y has the likelihood function (see Eq. (5.1-11))... 

Global Two-Stage Method. An extensive description of the method is provided by Steimer et al. The global two-stage (GTS) approach has been shown, through simulation, to provide unbiased estimates of the population mean parameters and their variance-covariance, whereas the estimates of the variances were upwardly biased if the STS approach was used. These simulations were done under the ideal situation that the residual error was normally distributed with a known variance. However, it is a well-known fact that the asymptotic covariance matrix used in the calculations is approximate and under less ideal conditions, the approximation can be poor. ... 

Normal distribution theory can be used to test whether a particular sample value is consistent with other values or with our past experience. If the mean p and the variance are known, then we can determine how deviant an observed value x,-, appears to be by calculating the statistic Z = (x,- - p)/cr and comparing this with the table ofstandard normal deviates. Suppose that one of the values for our QC specimen was 170ng/ml. Past experience has led us to believe that the results for this QC specimen are normally distributed with p = 207.6 ng/ml and cr = 14.1. Is the... 

The rank modification of von Neumann testing for data independence as described in Madansky (1988) and Bartels (1982) is applied. Although steady-state identification is not the original goal of this technique, it indicates if a time series has no time correlation and can thus be used to infer that there is only random noise added to a stationary behavior. In this test a ratio v is calculated from the time series, whose distribution is expected to be normal with known mean and standard deviation, in order to confirm the stationarity of a specific set of points. 

When d,j is independent of 5,y, it is known [8] that A( ) obeys a normal distribution with mean [(A — l)3 — (N — l)]/6 and variance (N — l)N2 (N — 2)2/36. Therefore, if the probability of A( j under this null hypothesis is small enough (e.g., less than 0.5%), we can reject this hypothesis and confirm that dij is related to 5,y. This means that object i is placed at a suitable location in the embedded space with a certain confidence level. 

The part-per-million concentration of a particular toxic in a wastewater stream is known to be normally distributed with mean, wj = 100 and a standard deviation (j = 2.0. Calculate the probability that the toxic concentration, C, is between 98 and 104. 

Note that the standard deviation can be used to compute the percentile rank associated with a given data point (if the mean and standard deviation of a normal distribution are known). In such a normal distribution, about 68% of the data points are within one standard deviation of the mean and about 95% of the data points are within two standard deviations of the mean. 

As a numerical illustration, it will be assumed that the standard deviation a of the Normal distribution is known, and the mean p of the distribution is to be estimated from the sample average, say X( ), from n measurements. The sample average A( ) is a random variable with a standard deviation equal to a/.Jn. The mean p) of the distribution is estimated by Eqn. 5.13 and is given by ... 

This example is actually unrealistic in two ways. First, we don t run trials with two experimental treatments unless they are different doses of the same treatment, which is not the case here and second, for reasons discussed in Chapter 3, we are generally interested in controlled comparisons and not population averages. However, to avoid some complications we shall accept the example as described here with the further simplification that we will assume that the variance of individual responses is known and the same for each treatment.) We suppose that the sample mean for a given treatment i is and is Normally distributed with variance erf /n. 

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