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
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
Z = ln(X).
Alternatively, when the random variable Z has a Normal
distribution, the random variable X has a Lognormal
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
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.
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