Estimates
the mean and the standard deviation,
μ, σ, of a (Gaussian) random variable
from sums of the values of unequal (or equal) size samples.
(The summands are, thus, assumed not known.)
A typical case are trucks of different sizes loaded with bags,
the bag being the item with random weight.
(The base data are inspired in real bags of fertilizers.)
Simulated values are used, from a Gaussian with
given parameters mean and standard deviation,
μ and σ. (Here, the values are weights of bags
or other additive values.)
There are T samples ("T" as in "truck"),
and the sample sizes are ni, i = 1..T,
in the given vector n.
(The samples are, indeed, commercial batches or lots.)
The estimation of μ and σ
through maximum likelihood gives:
Note that the last two formulas must be understood
as applying to each record, i.e., each set of
n (above) sums ("weighings"), as interests the practician.
For the simulation, each record contributes with its variance,
and from the average, a, of all these variances we finally get:
σ^ = √[N a ⁄(T − 1)].
(This is, of course, dimensionally homogeneous.)
The procedure estimates μ (which is trivial)
and σ from a set of experimental data
that are here simulated but which routinely exist in many industries.
The computation being lengthy, if the number of trials
is excessive (for the total lot sizes), it is reduced
(trials × N ≤ 108).
Draws a plot of f, the 'pdf', and F,
the 'cdf', of the variable simulated: weighted average
(around μ); or variance
(around σ²). |
