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Operations Research Models and Methods
Computation Section
Subunit Continuous Distributions
 - Lognormal

The Lognormal distribution exhibits a variety of shapes as illustrated in the figure. The random variable is restricted to positive values. Depending on the parameters, the distribution rapidly rises to its mode, and then declines slowly to become asymptotic to zero.

The figure shows three cases with the parameters and moments for each.


The Lognormal has two parameters that are the moments of the related Normal distribution.

The Lognormal and the Normal distributions are closely related. When some random variable X has a Lognormal distribution, the variable Z has a Normal distribution when

Z = ln(X).

Alternatively, when the random variable Z has a Normal distribution, the random variable X has a Lognormal distribution when

X = exp(Z).

We see in the figure below, the Normal distribution with mean 0 and standard deviation 1 is at the left. The corresponding Lognormal distribution is at the right.


For some cases, one might be given the mean and variance of the random variable X and would like to find the corresponding parameters of the distribution. Solving for the parameters in terms of the moments gives the expressions at the left. In the example below, we use statistical estimates of the mean and variance to compute the parameters of the distribution.



Example: Consider the material output from a rock crushing machine. Measurements have determined that the particles produced have a mean size of 2" with a standard deviation of 1". We plan to use a screen with 1" holes to filter out all particles smaller than 1". After shaking the screen repeatedly, what proportion of the particles will be passed through the screen? For analysis purposes we assume, the size of particles has a Lognormal distribution.

The given data provides estimates of the parameters of the distribution of X. The formulas at the left compute the values of the parameters of the Lognormal distribution. The spreadsheet segment computes the parameters and evaluates the required probability. Approximately 10% of the particles will pass through the screen.

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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved

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