Log Normal Law

For the determination of maximum likelihood estimators of logarithmic normal distribution parameters μ and σ we can use the same procedure as for normal distribution. Note that examples of variables that have roughly logarithmic normal distributions include the size of silver particles in a photographic emulsion, the survival time of bacteria in disinfectants, human weight and blood pressure, and the number of words written in George Bernard Shaw`s sentences. The term «log-normal» results from the logarithm on both sides: The mode of a logarithmic distribution at which the probability function takes its maximum value can be specified as follows: The characteristic function E [ e i t X ] {displaystyle operatorname {E} [e^{itX}]} is defined for real values of t, but is not defined for a complex value of t, which has a negative imaginary part, and therefore the characteristic function at the origin is not analytical. Therefore, the characteristic function of the logarithmic normal distribution cannot be represented as an infinite convergent series. [9] In particular, Taylor`s formal series differs: Fig. 1.20. Cumulative distribution of particle size in the lognormal probabilistic coordinate network constructed from the data in Table 1.4. For n finitely, these estimators are biased. While the distortion for μ^{displaystyle {widehat {mu }}} is negligible, a less biased estimator for σ {displaystyle sigma } as for the normal distribution is obtained by replacing the denominator n with n−1 in the equation for σ^2 {displaystyle {widehat {sigma }}^{2}}. There can be several reasons for using logarithmic normal distributions in conjunction with normal distributions. In general, most normal logarithmic distributions are the result of using the natural logarithm, where the basis is e = 2.718.

However, the normal logarithmic distribution can be scaled with a different basis, which affects the shape of the normal logarithmic distribution. One of the most common applications that use logarithmic normal distributions in finance is stock price analysis. The potential returns of a stock can be displayed graphically in a normal distribution. However, stock prices can be plotted in a logarithmic normal distribution. The logarithmic-normal distribution curve can therefore be used to better identify the composite return that the stock can expect over a given period. Finally, the variance of the logarithmic normal distribution is Var[X]=(eσ2−1)e2μ+σ2,text{Var}[X] = (e^{sigma^2}-1)e^{2mu+sigma^2},Var[X]=(eσ2−1)e2μ+σ2, which can also be written as (eσ2−1)m2big(e^{sigma^2}-1big)m^2(eσ2−1)m2, where mmm is the mean of the above distribution. Subsequently, these lognormal distributions can be used to sample the failure rate of the corresponding valve for each simulation in the Monte Carlo simulation method [Zio, 2013]. Subsequently, with N = 1000 simulations of the Monte Carlo method N, we determine the possible probabilities of occurrence of EL, which can be represented in a histogram, as shown in Fig. 5.27. Finally, this histogram can be used to estimate the expected value of the probability of occurrence of the EL and the desired confidence interval, taking into account the propagation of uncertainty.

For this example, the following estimates were obtained for one year of mission time: A probability distribution is not determined solely by the moments E[Xn] = enμ + 1/2n2σ2 for n ≥ 1. That is, there are other distributions with the same set of moments. [3] In fact, there is a whole family of distributions with the same moments as the normal logarithmic distribution. [ref. needed] The probability distribution function for a random variable is more conveniently specified as a standard distribution type with certain distribution parameters. Regressions of available observations of certain quantities do not always provide sufficient information to allow an interpretation of the type of distribution for the uncertain quantity. A selection of the type of distribution must be made. The results of a reliability analysis can be very sensitive to the end of the probability distribution. Therefore, a correct choice of distribution method is often crucial. Average and standard deviations are usually determined from detected data sources. The default lognormal distribution has a location setting of 0 and a scaling parameter of 1 (shown in blue in the screenshot below). If Θ = 0, the distribution is called a lognormal distribution with 2 parameters.

Some examples of lognormal density functions. Image: By Krishnavedala| Normal distributions of Wikimedia Commons can cause some problems that normal log distributions can solve. Primarily, normal distributions can allow negative random variables, while logarithmic normal distributions contain all positive variables. There is an interesting inequivalence between the log-normal description of electronic components and the function of increasing danger to human life [3,12]. For living species, internal aging processes apparently produce species-specific biological clocks with higher hazard functions than t → ∞. In addition, the presence of redundancy at many levels complicates the form of the hazard function. For example, the loss of a kidney does not lead to death in humans. In addition, random external variables, such as bacteria, viruses, accidents, cigarette, alcohol or drug addiction, become increasingly deadly as the body`s recuperative powers diminish. Unlike electronic devices, human longevity cannot be accurately predicted by physical examination.

To perform this uncertainty analysis, it is necessary to examine the data used to estimate failure rates, as shown in Table 5.5. As explained in point 5.5.1, a point estimate of the failure rate (λ^) and its 95th confidence interval can be made as a function of the number of failures and the total operating time of each valve. Taking into account the upper and lower bounds of this confidence interval and using equation (5.4), a lognormal distribution was used to model the uncertainty regarding the failure rate of each valve, as shown in the last two columns of Table 5.5. Let Z {displaystyle Z} be a normal standard variable, and μ {displaystyle mu } and σ > 0 {displaystyle sigma >0} two real numbers. Then the distribution of the random variable can be derived by leaving z = ln ( x ) − ( μ + n σ 2 ) σ {displaystyle z={tfrac {ln(x)-(mu +nsigma ^{2})}{sigma }}} inside the integral. However, the logarithmic normal distribution is not determined by its moments. [7] This implies that it cannot have a moment-generating function defined in a neighborhood of zero. [8] In fact, the expected value E [ e t X ] {displaystyle operatorname {E} [e^{tX}]} is not set to a positive value of the argument t {displaystyle t} because the defining integral diverges.