Normally Distributed Observations with Known Mean

5 NORMALLY DISTRIBUTED OBSERVATIONS WITH KNOWN MEAN 

In this section we will consider that we have normal observations where the mean /X is a known constant, and the variance is the unknown parameter. Again, the observations come from a one-dimensional exponential family so the analysis is simple. 

Single Draw from a Normal Distribution with Known Mean 

Suppose y is a random draw from a normal fi, r ) distribution where the mean /x is assumed to be a known constant. This is a draw from a one-dimensional exponential family with single parameter cr, since we can write the likelihood function as 

We see that this is a member of the one-dimensional exponential family with 

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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. 

Suppose 100 tosses were made and the number of heads was recorded, and the experiment is repeated many times. It would be observed that although usually there would be about 50 heads for every 100 tosses, occasionally there would be substantially greater or fewer. If the frequency of heads were plotted as a function of the number of heads observed in 100 tosses, a curve shape would be found that is entirely predictable. This shape, known as a normal distribution or normal curve, is shown in Fig. 2.4a. The primary virtue of a normal distribution is that because it is predictable, it can be described with two characteristic numbers, a mean value and a standard deviation. These are shown in Fig. 2.4a and are defined mathematically as... 

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... 

Thus, there are two parameters in the model. Fiuthermore, let m have a normal distribution with mean zero and variance and suppose that the true value of is known. This effectively makes om model have only one parameter, m. We can plug these models into the Bayes mle machinery, as embodied in Eq. (8.2) some algebraic calculations would reveal that the posterior distribution of m given the data has a closed-form solution. It is normal with mean nP-x/ nP- - - i ) and variance nf- + In the mean, the bar over the x denotes the average of the observations. 

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... 

When the population distributirai is known, for example X is normally distributed with mean p and variance o, we may calculate the probability of X being equal to or larger than x (X > x). We use that principle for e.g. process control by control charts. The production process has known targets for p and and when an uncommon value of variable X > x is observed, the production process should be adjusted. 

Example 1 A random sample yi,---, Vn is drawn from a distribution having an unknown mean and known standard deviatioru Usually, it is assumed the draws come from a normal p., a ) distribution. However, the statistician may think that for this data, the observation distribution is not nomuil, but from another distribution that is also symmetric aiul has heavier tails. For example, the statistician might decide to use the Laplace a, b) distribution with the same mean p and variance The mean arut variance of the Laplace a, b) distribution are given by a and 2b, respectively. The observation density is given by... 

Example 16 (continued) We continue with our example where we have a single random drawn from each of j = 1,..., J normally distributed populations where each population has its own mean and its own known variance. The observation yj comes from a normal pj, Oj) distribution with known variance aj. The population distributions are related. We model this relationship between the populations by considering the population means p, ..., pj to be random draws from a normal T, ip)... 

If the total population is known and, therefore, also the true mean fi and the standard deviation A, to infer the value that corresponds to a given percent of survival is rather an easy game. Assuming that a normal distribution holds, what shall be done is to evaluate the number k of standard deviation A to subtract to the true mean ji. But when the true mean is not known because the population of data is to large with respect to the sample size (see Eq. 4.1) and the mean available x is just the sample mean, the question arises as to how close or far we really are from the true one. The question can be answered only in terms of confidence interval C. In general terms, if a population parameter is not known, for instance the true mean it can always be estimated using observed sample data. Estimated actually means that its value will never be exactly determined, but it may be included in a range of values whose size depends on the confidence we want to know it. As the... 

The well-known SD unit or normal equivalent deviate is such a measure. It is calculated as the difference between the observed value and the mean of the reference values divided by their standard deviation. Several similar ratios have been suggested/ but none has significant advantages over the other. All produce very confusing values if the reference distribution is very skewed. An observed value (e.g., with an SD unit of 2.2) would be above the 97,5 percentile if the reference distribution has a Gaussian shape, but might be well below the upper reference limit of a positively skewed distribution. Mathematical transformation of the reference distribution to the Gaussian shape may solve this problem. ... 

It is instructive to think about the meaning of the prior distribution in order to understand why this happens. Having a Normal prior distribution for a true population mean and then observing a sample from that population is formally analogous to the following. One has observed an infinity of population means (these are what make the prior) one has drawn a population at random from this infinity (but one does not know which one), observed a sample from it and now proceeds to make inferences about the mean. Thus implicitly, one is comparing the sample mean with an infinity of known population means. Thus the comparison with some locally observed sample means adds no further useful information and it is therefore also irrelevant whether the sample mean is the largest or not in this sample. 

When the observations come from a normal p,a ) distribution where the mean /x is known, the conjugate prior for is S times an inverse chi-squared distribution with degrees of freedom. The posterior is S times an inverse chi-squared distribution with k degrees of freedom where... 

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