Distribution lognormal

The varianee for any set of data ean be ealeulated without referenee to the prior distribution as diseussed in Appendix I. It follows that the varianee equation is also independent of a prior distribution. Here it is assumed that in all the eases the output funetion is adequately represented by the Normal distribution when the random variables involved are all represented by the Normal distribution. The assumption that the output funetion is robustly Normal in all eases does not strietly apply, partieularly when variables are in eertain eombination or when the Lognormal distribution is used. See Haugen (1980), Shigley and Misehke (1996) and Siddal (1983) for guidanee on using the varianee equation. 

Analytieal solutions to equation 4.32 for a single load applieation are available for eertain eombinations of distributions. These coupling equations (so ealled beeause they eouple the distributional terms for both loading stress and material strength) apply to two eommon eases. First, when both the stress and strength follow the Normal distribution (equation 4.38), and seeondly when stress and strength ean be eharaeterized by the Lognormal distribution (equation 4.39). 

B. Equivalent mean (/x) and standard deviation a) Lognormal distribution... 

Lognormal distribution (integrate using numerical techniques or use SND table)... 

If air quality data at a receptor for any one averaging time are lognormally distributed, these data will plot as a straight line on log probability graph paper (Fig. 4-9) which bears a note Sg = 2.35. Sg is the standard geometric deviation about the geometric mean (the geometric mean is the Nth root of the product of the n values of the individual measurements). 

Larsen (18-21) has developed averaging time models for use in analysis and interpretation of air quality data. For urban areas where concentrations for a given averaging time tend to be lognormally distributed, that is, where a plot of the log of concentration versus the cumulative frequency of occurrence on a normal frequency distribution scale is nearly linear,... 

The Reactor Safety Study extensively used the lognormal distribution (equation 2.5-6) to represent the variability in failure rates. If plotted on logarithmic graph paper, the lopnormal distribution is normally distributed. 

WASH-1400 treated the probability of failure with time as being exponentially distributed with constant X. It treated X itself as being lognormally distributed. There are better a priori reasons... 

A type of time dependence that is available in most codes evaluates the exponential distribution at specified times. This is the constant failure rate - constant repair rate approximation ( Section 2.5.2). This may not be realistic as indicated by Figure 2.5-2 in which the failure rate is not constant. Furthermore, Lapides (1976) shows that repair rates are not constant but in many casc. appear to be lognormally distributed. 

Loss of offsite power at nuclear power plants is addressed in EPRI NP-2301, 1982 giving data on the frequency of offsite power loss and subsequent recoveiy at nuclear power plants. Data analysis includes point estimate frequency with confidence limits, assuming a constant rate of occurrence. Recovery time is analyzed with a lognormal distribution for the time to recover. 

It is assumed that both and u are lognormally distributed with logarithmic standard deviations of and P, j, respectively. The advantages of this formulation are ... 

The product form of equation 5.1-6, and its assumed lognormal-distribution make the fragility computations mathematically tractable. 

The justification for the use of the lognormal is the modified Central Limit Theorem (Section 2.5.2.5). However, if the lognormal distribution is used for estimating the very low failure frequencies associated with the tails of the distribution, this approach is conservative because the low-frequency tails of the lognormal distribution generally extend farther from the median than the actual structural resistance or response data can extend. 

Lognormal distribution Similar to a normal distribution. However, the logarithms of the values of the random variables are normally distributed. Typical applications are metal fatigue, electrical insulation life, time-to-repair data, continuous process (i.e., chemical processes) failure and repair data. 

Error factor The ratio of the 95th percentile value to the median value of a lognormal distribution. 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

If the variability (a) depends on concentration, prior knowledge of concentration may be required to use these formulas. If the relative standard deviation (RSD) is constant with respect to concentration, then the formulas can be applied by interpreting a and E as relative standard deviation and relative error, respectively. A common case in which RSD is constant with respect to concentration is when analytical results are lognormally distributed. For example, suppose it is desirable to estimate the average concentration with 95% confidence that the estimate will be within 10% of the true value if the relative standard deviation is 25%. Then... 

Means and standard deviations for these distributions were normalized to daily breathing rates (m3/day), and an acceptable range was defined. It was assumed that the "day" represents the duration of time within a working day that chlorpyrifos may be handled by an individual (0.25 to 6.0 hr). It was also assumed that exposures would be negligible for the remainder of the working day following application or other contact. Both the dermal and inhalation exposures were assumed to follow lognormal distributions, which is consistent with common practice for exposure data distributions (for example, in the Pesticide Handlers Exposure Database, PHED). 

The Log-Probit Model. The log-probit model has been utilized widely in the risk assessment literature, although it has no physiological justification. It was first proposed by Mantel and Bryan, and has been found to provide a good fit with a considerable amount of empirical data (10). The model rests on the assumption that the susceptibility of a population or organisms to a carcinogen has a lognormal distribution with respect to dose, i.e., the logarithm of the dose will produce a positive response if normally distributed. The functional form of the model is ... 

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