Lognormal distribution probability density function

Figure 4.3 Shapes of the probability density function (PDF) for the (a) normal, (b) lognormal and (c) Weibull distributions with varying parameters (adapted from Carter, 1986)...

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). 

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

The relative velocity between the liquid and the gas is considered to be one of the most important factors that affect the liquid breakup process during gas atomization. For a given gas nozzle design, particle size is controlled by the atomizing media pressure and melt flow rate. The droplet size distribution for various gas-atomized alloys has been reported generally to foUow a lognormal distribution [13-17]. Two numbers d o, median mass diameter, and ffg, geometric standard deviation, are usually used to describe the entire size distribution. The mass probability density function, p(d), of the droplet-size distribution can be expressed by [18-20] ... 

The statistical distributions used in maintainability studies are characterized by a restore probability density function [g(t)j. Usually, the main distributions used are WeibuU, Exponential, Lognormal or Normal distribution. 

The lognormal distribution describes the long-term variation of the received envelope. The corresponding probability density function is given by... 

Define the received power as w=r l, where r is the received signal envelope. Let p(W) be the probability density function of the power W, where W = 10 log w. In a lognormal environment r has a lognormal distribution and... 

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... 

Clearly, any probability density function and corresponding cumulative probability distribution could be used to describe the uncertainty in the data. Trapezoidal, normal, lognormal, and so on, are used routinely to describe uncertainty in data. However, for simplicity, the following discussions are confined to triangular distributions. The eight-step method for quantifying uncertainty in profitability analysis is illustrated next. 

The probability density functions for the normal and lognormal distributions are given in eqs. (12-8) and (12-9), respectively ... 

If, for example, F(s) is taken as a cumulate lognormal distribution function, dF(s)lds is the lognormal probability density function, i.e.,... 

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... 

If the ciunulated probability depends on a physical size that is by definition only positive and increases with the physical size, in our case the overpressure and the blat impulse, the lognormal density and cumulated distribution function of (9) is the right choice. In this case the lognormal density for all physical sizes smaller than z are used to compute the cumulated probability F+ (z). This can be interpreted as considering all damage cases, also for smaller z-values up to the actual z-value. 

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