How to Compute Aggregate Best and Worst Cases for a Large Number of Estimates?

Question

I need to aggregate multiple estimations, but I haven't been able to find a built-in function in Mathematica that aggregates multiple probablity estimations (I am specially interested in estimations based on the PERTDistribution)

I am being told that I need to:

1. Compute the standard deviation of each estimation
2. Compute the square of each task’s standard deviation, which is known as the variance.
3. Total the variances.
4. Take the square root of the total.

First of all, is that correct? And if it is, is there a function in Mathematica that can do all that for me? Something like a hypotetical AggregateDistribution that would work like:

AggregateDistribution[PERTDistribution[{min, max}, mode], numberOfEstimatesToAggregate]

Any guidance on a function that provides this or a way to do it would be welcome.

Answer 1

A different approach which only traverses the samples once and efficiently wraps each step around the result from the previous stage using ComposeList. It just about falls into the class of one line answers.

sampleSize = 100;
numOfSamples = 10;

ComposeList[{#^2 &, Total, Sqrt}, 
StandardDeviation/@RandomVariate[PERTDistribution[{0, 1}, 0.5],{numOfSamples, sampleSize}]]

Answer 2

If I follow your question, you want to generate a number of samples of data from a PERTDistribution then compute some statistics of those samples. Something like this should work:

sampleSize = 100;
numOfSamples = 10;
data = Table[RandomVariate[PERTDistribution[{0, 1}, 0.5], sampleSize], {i, numOfSamples}];
std = StandardDeviation[#] & /@ data
var = Variance[#] & /@ data
varTotal = Total@var
varTotalSqrt = Sqrt[varTotal]

And as a function incorporating (now both of) Mr. Wizard's suggestions:

sampleSize = 100;
numOfSamples = 10;

aggregateDistribution[sampleSize_, numOfSamples_] := 
 Module[{data, std, var, varTotal, varTotalSqrt},
  data = RandomVariate[PERTDistribution[{0, 1}, 0.5], {sampleSize, numOfSamples}];
  std = StandardDeviation /@ data;
  var = Variance /@ data;
  varTotal = Total@var;
  varTotalSqrt = Sqrt[varTotal];
  {std, var, varTotal, varTotalSqrt}
  ]

aggregateDistribution[sampleSize, numOfSamples] // ColumnForm

Answer 3

I hope that this response is sufficiently different in spirit that it deserves a separate answer.

Mathematica's rich set of functionality extends into direct manipulation of statistical distributions. We can leverage this analytic capability to avoid the need for large numbers of repeated sample trials and work directly with the distributions themselves. This is particularly useful for distributions whose statistical quanities are different from the Normal distribution.

One of which is the Beta family of distributions, of which the PERT is a member.

Using the Mathematica function Variance on a sample drawn from a distribution other than the normal may lead to a false result, this is the case with the PERT.

We can get directly:

dist=PERTDistribution[{0.25, 5}, 1];

{StandardDeviation[dist], Variance[dist]}

{0.79884, 0.638145}

Showing that the standard deviation and variance are related in a very different way from that in the Normal distribution.

I'm not certain what totaling a set of variances from a non Normal distribution and then finding the square root of them is designed to achieve.