Lognormal random variable

Lognormal Random Variable. Every normally distributed random variable, y, is uniquely associated with a lognormally distributed random variable, x, whose probability density function is completely characterized by its geometric mean, GM, and geometric standard deviation, GSD (2). 

The cumulative distribution function for the lognormal random variable x is derived by combining Equations A-4 and A-5. 

Fig. 4.4 Approximation of a lognormal random variable with <7y = 0.1 performing an lEE...

Leipnik RB (1991) On Lognormal Random Variables I-The Characteristic Function. Journal of Australian Mathematical Society 32 327-347. 

It is obvious that the failure probability calculated using the FOSM approach is accurate only in two cases (1) when G is a linear function of X and the quantities in AT are statistically independent normal random variables or (2) when G is a multiplicative function of X and the quantities in X are statistically independent lognormal random variables. In many practical examples, it is unlikely that all the variables are statistically independent normals or lognormals. Nor it is likely that G is an additive or multiplicative function of X. In such cases, Eq. 24 is oifiy approximate however, it can be used to provide a rough idea of the level of risk or reliability. 

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. 

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. 

The effect due to specimen differences is assumed to be a random variable having a lognormal distribution. The assumption of a lognormal distribution implies that the logarithm (base 10) of the random variable has a normal distribution with some mean p and variance [i.e., log X V N(p, o )]. Here we assume that... 

In this method, each assessment factor is considered uncertain and characterized as a random variable with a lognormal distribution with a GM and a GSD. Propagation of the uncertainty can then be evaluated using Monte Carlo simulation (a repeated random sampling from the distribution of values for each of the parameters in a calculation to derive a distribution of estimates in the population), yielding a distribution of the overall assessment factor. This method requires characterization of the distribution of each assessment factor and of possible correlations between them. As a first approach, it can be assumed that all factors are independent, which in fact is not correct. 

Coalescence Growth Mechanism. Following the very early step of the growth represented by Eq. (1), many nuclei exist in the growth zone. Hence Eq. (2) would be a major step for the crystal growth. Since there are many nuclei and embryos with various sizes in the zone, Uy in Eq. (2) can be assumed to be a random variable. Due to mathematical statistics, the fraction of volume approaches a Gaussian after many coalescence steps (3). A lognormal distribution function is defined by... 

Separating Equation A-10 into two equations helps to clarify the procedure needed to quickly calculate the q1 quantile, Xq, of pdf(x) when x is a lognormally distributed random variable. Since q and Zq are related by Equation A-2, Equation A-ll can be used to solve for Xq given q or for q given Xq. 

The Effect of Imperfect Sampling. Let St be the random variable which estimates the true value of the lognormally distributed random variable (x). The value of (St) is determined by sampling (x) and analyzing with a process which has inherent uncertainty associated with it. The uncertainty is described by the coefficient of variation of the analysis, CV. If CV < 0.3, then %) can be modelled adequately as a lognormally distributed random varible characterized by GM and GSV as defined below (1). 

A probability distribution is a mathematical description of a function that relates probabilities with specified intervals of a continuous quantity, or values of a discrete quantity, for a random variable. Probability distribution models can be non-parametric or parametric. A non-parametric probability distribution can be described by rank ordering continuous values and estimating the empirical cumulative probability associated with each. Parametric probability distribution models can be fit to data sets by estimating their parameter values based upon the data. The adequacy of the parametric probability distribution models as descriptors of the data can be evaluated using goodness-of-fit techniques. Distributions such as normal, lognormal and others are examples of parametric probability distribution models. 

Suppose the expression data ykji are generated from a random variable Ykj that follows a lognormal distribution with parameters r]kj and a so that the mean Ukj is enki+ahl2 and the variance is - 1)- In particular, Xkj = log(Ykj) is... 

Different safety factors may have been used in the derivation of the reference values of the individual substances (RfDA deterministic HI thus sums risk ratios that may reflect different percentile values of a risk probability distribution. Assessment and interpretation of the uncertainty in the HI may be severely hampered by this summation of dissimilar distribution parameters. In a probabilistic risk assessment, the uncertainty in the exposure and reference values is often characterized by lognormal distributions. The ratio of 2 lognormal distributions also is a lognormal distribution. The variance in a quotient of 2 random variables can be approximated as follows (Mood et al. 1974, p 181) ... 

This is a lognormal distribution, it is the probability density function of the random variable p, characterizing the relative filling of the cells. 

The stochastic dynamics usually applied in finance literature is generated by lognormal- or close-to-lognormal -distributed random variables. Leip-nik (1991) shows that the series expansion of order M of a Gog) characteristic function in terms of the cumulants diverges for M oo. Hence, the... 

We show that the application of the EE is admissible leading to accurate results, even in the case of lognormal-distributed random variables. This good-natured behavior of the EE, firstly comes from the fact that the volatility t5 ically occurring in bond markets is rather low, generating more close-to-normal -distributed random variables. Secondly, the series expansion of the (log) characteristic function in terms of the cumulants can be practically applied for M lower than a critical order Me. 

Given the mean /ij, and the standard deviation of a normal-distributed random variable y we obtain the pdf of a lognormal-distributed random variable X by... 

Together with the one-to-one mapping between the cumulants and the moments (3.2) we can compute the cumulants of a lognormal-distributed random variable x as follows... 

As in chapter (3), where we approximated the density function of a and lognormal-distributed random variable, we again focus in our analysis on all parameters K, such that T1 [AT] > 10 holds. This implies that we are able to analyze the accuracy of this method, even for far out-of-the-money options. 

Then, plugging the cumulants (3.15) of a lognormal-distributed random variable z in the lEE scheme leads to the following series expansion of the probability that z exceeds the strike price K... 

E.g. a lognormal-distributed random variable with cTy = 0.1 and jity = 0... 

Finally, it can be pointed out that the new lEE is an excellent method for the approximation of exercise probabilities, even if the underlying random variable is lognormal- or lognormal-like distributed and the pdf does not exist in closed-form. Flence, the lEE ean be applied for the approximation of the single exercise probabilities given an underlying random variable which is composed by a sum of multiple lognormal-distributed random variables. 

It is well known that the product of lognormal-distributed random variables is lognormal. Hence, in contrast to the transform Et z) for z G C it is possible to derive a closed-form solutions for the transform Et n) = Ptfi n) with n G hm. This implies that we are able to compute the moments of V To Ti ) in closed-form and the single exercise probabilities can be approximated by performing an lEE. 

Let us assume they represent N = 7 independent measurements (this is often not the case, since values from the same source are quoted in several references). Then the explosion limit may be assumed to be a random variable, i.e. a variable which adopts certain values with a certain probability. Random variables are described by probability distributions (vid. Appendix C). In what follows the logarithmic normal (lognormal) distribution (vid. Sect. 9.3.4) is used to represent the values... 

The probability distributions of concrete column resistance Rn or Rm and permanent action effect Nq or Mg are close to a normal distribution (Ellingwood, 1981 ISO 2394 1998 JCSS2000). The distribution of the safety margins is close to lognormal but they may be treated as a normal random variables (Swed et al. 2005 Sugiyama Yoshida 2008). 

The probabilistic models apphed in the reliability analysis are listed in Table 3. Some of the basic variables entering expression (11) are assumed to be deterministic values denoted DET (reinforcement area, some geometric characteristic, coefficients kt and k2) while the others are considered as random variables having normal (N), lognormal (LN), Beta (BET) and Giunbel (GUM) distributions. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

The time to failure of specimens subjected to an accelerated lifetest is a random variable which is usually distributed according to a lognormal cumulative probability function (CPF). Once the shape and the values of the parameters characterising the CPF have been determined, it is possible to predict the failure rate of the sampled population as a function of the time in service." ... 

Usually in reliability theory (of technical system) probability distributions of random variable T (time-to-failure) are exponential Exp(k) (in case of constant failure rate), Weibull W k, T ) (with appropriate choice of parameters a variety of failure rate behaviors can be modeled), lognormal Log-N i, a ), gamma E(a, p), etc. (Lewis, 1994, Zio, 2(X)7). 

The PSA approach to the evaluation of probabilistic pressure capacity involves limit state analyses. The limit states represent the ensemble of possible failure modes of the confinement functions. In principle, containment failure may be interpreted as incipient leakage (which may be a small controlled le ) or a large catastrophic break. The median capacity of a given failure mode is typically dependent on several factors, including materid properties, modelling assumption and failure criteria. Considerable uncertainty and variability may be introduced in these factors and should be properly accounted for in the probabilistic overpressure fragility evaluation. To incorporate the uncertainty and variability in the formulation, the pressure capacity for each failure mode is represented as a random variable with a lognormal distribution. 

After the selection of the analytical model, the next step is to generate the structural simulations. Due to the probabilistic nature of seismic fragility analysis, some of the major structural parameters within the analytical model are considered as random variables with appropriate probability density functions assigned to them. Normal or lognormal distributions are commonly used for convenience. These can be mechanical properties like stiffness or strength to account for the material variability or geometric properties like... 

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