## Probabilistics 2

### “Probabilistics” 2

*There is more to Monte Carlo simulation than replacing constants with probability densities.*

#### The Data:

In the mid-1970s Dennis Virkler, then a Ph.D. student of Professor Ben Hillberry at Purdue, conducted 68 crack growth tests of 2024-T3 aluminum. These tests were unusual because they were conducted expressly to observe random behavior in fatigue. While most crack growth tests measure crack length after some number of cycles, Virkler measured cycle count at 164 specific crack lengths. This provided a direct measure of variability in cycles, rather than the usually observed variability in crack length at arbitrary cyclic intervals.

#### Conventional Monte Carlo Simulation:

Unlike many engineering analytical results, probability estimates are difficult to verify experimentally. This unfortunate reality has perpetuated the misuse of a valid statistical tool, and the consequences may not be apparent for years to come.

Most engineering Monte Carlo simulations are performed this way.

1. Set up a conventional deterministic analysis;

2. Replace constants with probability distributions;

3. Sample once from each distribution;

4. Compute the deterministic result and store the answer;

5. Repeat steps 3 and 4 many times;

6. Compute the mean and standard deviation of the collected results.

Sadly, many engineers are unfamiliar with the implicit statistical assumptions that are at the foundation of Monte Carlo simulation, but has been observed elsewhere “Simply not understanding the nature of the assumptions being made does not mean that they do not exist.”