Mean and variance

Let s consider the following problem. Two sets of blood samples have been collected from a patient receiving medication to lower her concentration of blood glucose. One set of samples was drawn immediately before the medication was administered the second set was taken several hours later. The samples are analyzed and their respective means and variances reported. ITow do we decide if the medication was successful in lowering the patient s concentration of blood glucose ... 

To begin, we calculate the global mean and variance and the means for each analyst. These results are... 

For specified probability and density functions, the respective means and variances are defined by the following ... 

In order to compare populations based on their respective samples, it is necessaiy to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effecd, the t statistic characterizes the way in which two sample means from two separate populations will tend to vaiy by chance alone when the population means and variances are equal. Consider the following ... 

The normal model can take a variety of forms depending on the choice of noninformative or infonnative prior distributions and on whether the variance is assumed to be a constant or is given its own prior distribution. And of course, the data could represent a single variable or could be multidimensional. Rather than describing each of the possible combinations, I give only the univariate normal case with informative priors on both the mean and variance. In this case, the likelihood for data y given the values of the parameters that comprise 6, J. (the mean), and G (the variance) is given by the familiar exponential... 

Two types of boundary conditions are considered, the closed vessel and the open vessel. The closed vessel (Figure 8-36) is one in which the inlet and outlet streams are completely mixed and dispersion occurs between the terminals. Piston flow prevails in both inlet and outlet piping. For this type of system, the analytic expression for the E-curve is not available. However, van der Laan [22] determined its mean and variance as... 

Table 8-8 summarizes the results of the cases discussed above including the boundary conditions, the expression for C(6) at z = 1, and the mean and variance for C(6). 

The Central Limit Theorem gives an a priori reason for why things tend to be normally distributed. It says the sum of a large number of independent random distributions having finite means and variances is normally distributed. Furthermore, the mean of the resulting distribution the sum of the individual means the combined variance is the sum of the individual variance.. ... 

Table 2.5-2 provides a convenient summary of distributions, means and variances used in reliability analysis. This table also introduces a new property called the generating function (M,0). 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

To deal with all kinds of normal distributions of different means and variances, the cumulative distribution is further normalized. This introduces a new variable u = x - ix)/(t. This operation changes a N(p, a) distribution to a N(0, 1) distribution. From Eq. (12.3) the following is obtained ... 

Tlie mean )i and tlie variance a" of a random variable are constants cliaracterizing die random variable s average value and dispersion about its mean. The mean and variance can be derived from die pdf of the random variable. If die pdf is miknown, however, the mean and die variance can be estimated on die basis of a random sample of observations on die random variable. Let X, Xj,. X, denote a random sample of n observations on X. 

A die is loaded so diat die probability of any face turning up is directly proportional to die number of dots on die face. Let X denote the outcome of tlirowing die die once. Find die mean and variance of X. 

The mean and variance of a random variable X having a log-normal distribution are given by... 

The random force is taken from a Gaussian distribution with zero mean and variance... 

A few examples should serve to illustrate the technique of calculating means and variances, and to point out the need for caution when using... 

Both the mean and variance of the Poisson distribution are equal to the parameter A. 

The last example brings out very clearly that knowledge of only the mean and variance of a distribution is often not sufficient to tell us much about the shape of the probability density function. In order to partially alleviate this difficulty, one sometimes tries to specify additional parameters or attributes of the distribution. One of the most important of these is the notion of the modality of the distribution, which is defined to be the number of distinct maxima of the probability density function. The usefulness of this concept is brought out by the observation that a unimodal distribution (such as the gaussian) will tend to have its area concentrated about the location of the maximum, thus guaranteeing that the mean and variance will be fairly reasdnable measures of the center and spread of the distribution. Conversely, if it is known that a distribution is multimodal (has more than one... 

This result checks with our earlier calculation of the moments of the gaussian distribution, Eq. (3-66). The characteristic function of a gaussian random variable having an arbitrary mean and variance can be calculated either directly or else by means of the method outlined in the next paragraph. 

Equation (3-88) enables us to calculate the characteristic function of the unnormalized random variable from a knowledge of the characteristic function of . For example, the characteristic function of a gaussian random variable having arbitrary mean and variance can be written down immediately by combining Eqs. (3-83) and (3-88)... 

A table giving the characteristic function, mean, and variance of some important probability distributions appears in Fig. 3-6. 

Fig. 3-6. Some Typical Characteristic Functions, Means, and Variances.

The mean and variance of the sum sn are related to the means and variances of the summands by means of the formulas... 

The Central Limit Theorem.—If 4>i,4>a, we identically distributed, statistically independent random variables having finite mean and variance, then... 

A more refined argument shows that the gaussian character of the sum in Eq. (3-279) is preserved in the limit as n - oo and Art - 0 so that we may conclude that Y(t) is exactly gaussian. Since a (one-dimensional) gaussian distribution is completely specified by its mean and variance, we need merely calculate [F(t)] and E[Y2(t)] to completely determinepr(y). This is easily done as follows ... 

Find the y that minimizes this integral, subject to the constraints that y is a density function with given mean and variance, i.e.,... 

I will present here the properties of various sources when the random variable considered is the field intensity. In this case, one has access to the mean and variance via a simple photodetector. The autocorrelation function can be interpreted as the probability of detecting one photon at time t + t when one photon has been detected at time t. The measurement is done using a pair of photodetectors in a start stop arrangement (Kimble et al., 1977). The system is usually considered stationary so that the autocorrelation function, which is denoted depends only on r and is defined by ... 

The mean and variance of the difference between B colla (from table) and B colu is determined for all 14 diets for each trial combination of dp, dpi and to, and the best values for dp, dpj and w chosen to minimize both the mean and the variance. These values turn out to be dp = +5, dN = +2 and CO = -0.75. Figure All.l shows a plot of the difference between the estimated and calculated collagen values for each diet for this particular DIFF, and it can be seen that, except for one point, the others are correctly estimated to within 1 or 1.5%o. 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

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