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					                                 Lognormal distribution
The generative process leading to a lognormal distribution was explained by Gibrat (1930) and
called "the law of proportional effect". It is also closely related to the power law distribution
(Zipf, Pareto). The lognormal distribution is based on the normal distribution. It is one of the life
time distributions. It is valid for values of X which are greater than zero. A lognormal
distribution is a continuous distribution of a random variable whose logarithm is normally
distributed. It is sometimes also called the Galton distribution.
 X is log normally distributed if Y=ln X is normally distributed. The Probability density function
of lognormal distribution is given as follows:
                                                      1  ln x   
                                           1                     
                                                      2  2 
                                f(x) =           e                     0<x< 
                                         x 2

It has two parameters  and  2 . The lognormal distribution is implemented by the Lognormal
Distribution class. It has two constructors. The constructor has two parameters. The first
parameter is the location parameter, and corresponds to the mean of the associated normal
distribution. The second parameter is the scale parameter and corresponds to the standard
deviation of the associated normal distribution. The location parameter is one that determines
the location or shift of the distribution. The scale parameter tells the spread of the distribution.
If scale parameter is large then the distribution will be more spread out. . Lognormal
distribution is skewed to the right.
The lifetime of a product that degrades over time is often modeled by a lognormal random
variable. For example this is a common distribution for the lifetime of a semiconductor laser.
Other distributions can also be used for this type of application. However, because the
lognormal distribution is derived from a simple exponential function of a normal random
variable, it is easy to understand and easy to evaluate probabilities.


Properties:
                                   Notation =

                            Parameters = σ2 > 0 — shape (real),
                                             μ ∈ R — log-scale
                                                                  x ∈ (0, +∞)




                              CDF =

                                      Mean =

                               Variance =

                                        Median         =
                                        Mode       =
                            Skewness           =

Relation to the log-normal distribution with other distribution:
Note that the Pareto distribution and log-normal distribution are alternative distributions for
describing the same types of quantities. One of the connections between the two is that they are
both the distributions of the exponential of random variables distributed according to other
common distributions, respectively the exponential distribution and normal distribution. (Both of
these latter two distributions are "basic" in the sense that the logarithms of their density functions
are linear and quadratic, respectively, functions of the observed values.)

      For sample sizes n = 3(1)99 and 100(10)1000, the approximate relationships between the
       Weibull β and Lognormal ρ were derived by a trial and error method. These non-linear
       relationships, equations (14) and (15), are functions of the sample size n. In order to see if the

       equations fit the data Let                              be independent log-normally distributed
       variables with possibly varying σ and μ parameters, and                    . The distribution of Y
       has no closed-form expression, but can be reasonably approximated by another log-normal
       distribution Z at the right tail. Its probability density function at the neighborhood of 0 is
       characterized in (Gao et al., 2009) and it does not resemble any log-normal distribution. A
       commonly used approximation (due to Fenton and Wilkinson) is obtained by matching the mean
       and variance:
      Lognormal distribution is a special case of semi-bounded Johnson distribution
      The same ranking procedure as was used for the Weibull distribution was also used for the
       Lognormal distribution.
      onte Carlo study (100 replications) was made.




Lognormal / lognormal: If X1 and X2 are lognormal random variables with parameters (μ1, σ12)
and (μ2, σ22) respectively, then X1 X2 is a lognormal random variable with parameters (μ1 + μ2,
σ12 + σ22).

Normal / lognormal: If X is a normal (μ, σ2) random variable then eX is a lognormal (μ, σ2)
random variable. Conversely, if X is a lognormal (μ, σ2) random variable then log X is a normal
(μ, σ2) random variable.

     Reliability and failure data, both from life testing and from in-service records, are often modeled
by the Weibull or Lognormal distributions so as to be able to interpolate and/or extrapolate results. As
the Weibull and Lognormal distributions are not from the same mathematical family, we are unable to
derive mathematical relationships between their shape and scale parameters directly. This study tries
to use a simulation method to determine their approximate relationship. The simulation was carried
out using WeibullSMITH™ software which uses the Monte Carlo method to generate samples from the
Weibull (or the Lognormal) distribution and can then fit the Lognormal (or the Weibull) line to the
sample points. The paper found approximate equations of shape parameters (or scale parameters) for
complete sample sizes varies from 3 to 99 and from 100 to 1000 between both distributions. For η=1,
β=0.5, 1, 3, 5 and n=10, 25, 50,100, the residuals between true ρ (shape parameter of the Lognormal)
and estimated ρ are less than ±0.003. Again in the same conditions as above, the residuals between
true θ (scale parameter of the Lognormal) and estimated θ are less than ±0.0005. The approximate
equations are simple, direct, and their accuracy is acceptable. They are, therefore, recommended for
use.



THE RELATIONSHIP BETWEEN THE SHAPE PARAMETERS OF THE WEIBULL AND LOGNORMAL
DISTRIBUTIONS

Since the Weibull and Lognormal distributions are not from the same mathematical family, we are
unable to derive mathematical relationships between their shape and scale parameters directly.
However, we can use a simulation method to determine their approximate relationship. The simulation
was carried out using WeibullSMITH software (Fulton, 2005) which uses the Monte Carlo method to
generate samples from the Weibull (or the Lognormal) distribution and can then fit the Lognormal (or
the Weibull) line to the sample points. For example, 25 random samples were generated from W(1, 1)
and the resulting data were fitted to both the Weibull and Lognormal lines using MRR as shown in
Figures 6 and 7 respectively. As we can see from these two graphs, W(1, 1) corresponds to L(0.577,
0.822) where the 0.822 is the value of ρ, the reciprocal of σ, and muAL= θ and sdF=eσ =3.377. Therefore,
the approximate relationships between the shape parameters and scale parameters of the Weibull and
Lognormal distributions can be determined.




Applications of Lognormal distribution

In biology, variables whose logarithms tend to have a normal distribution include:

1) Measures of size of living tissue (length, skin area, weight)
2) The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the
   direction of growth.
3) Certain physiological measurements, such as blood pressure of adult humans (after
   separation on male/female subpopulations)
    Consequently, reference ranges for measurements in healthy individuals are more accurately
estimated by assuming a log-normal distribution than by assuming a symmetric distribution
about the mean.

4) The log-normal distribution is used to analyze extreme values of such variables as monthly
   and annual maximum values of daily rainfall and river discharge volumes.
5) The distribution of city sizes is lognormal. This follows from Gibrat's law of proportionate
   (or scale-free) growth. Irrespective of their size, all cities follow the same stochastic growth
   process. As a result, the logarithm of city size is normally distributed. There is also evidence
   of lognormality in the firm size distribution and of Gibrat's law.
6) In reliability analysis, the lognormal distribution is often used to model times to repair a
   maintainable system.
Applications of lognormal Distribution
   1. Geology and mining
In the Earth’s crust, the concentration of elements and their radioactivity usually follow a
lognormal distribution.

  2. Environment and Atmospheric sciences
The distribution of particles, chemicals, and organisms in the environment is often log-normal.
For example, the amounts of rain falling from seeded and unseeded clouds differed significantly.
The atmosphere is a major part of life support systems, and many atmospheric physical and
chemical properties follow a log-normal distribution

   3. Food technology
Various applications of the log-normal distribution are related to the characterization of
structures in food technology and food process engineering. Such disperse structures may be the
size and frequency of particles, droplets, and bubbles that are generated in dispersing processes,
or they may be the pores in filtering membranes.

  4. Plant physiology
Recently, convincing evidence was presented from plant physiology indicating that the log-
normal distribution fits well to permeability and to solute mobility in plant cuticles.

   5. Phytomedicine and microbiology
Examples from microbiology and phytomedicine include the distribution of sensitivity to
fungicides in populations

   6. Atmospheric sciences and aerobiology
Another component of air quality is its content of microorganisms, which was—not
surprisingly—much higher and less variable in the air.

				
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Description: Lognormal distribution