J.D. Solomon

# Three Solutions for Establishing Useful Probability Distributions

Monte Carlo simulations are the best approach for assessing uncertainty for future infrastructure and facilities funding requirements. However, developing (or choosing) a probability distribution is a major factor in most technical professionals’ decisions not to use this approach. Here are three solutions for establishing a useful probability distribution.

**Probability Distributions: Definitions That Confuse**

A probability *density* function provides a relative likelihood that a given value would *equal *a given point in the sample space. The probability density function (PDF) is applied to continuous random variables, whereas the probability *distribution* function is defined for *discrete* random variables. For discussion purposes, these are often interchangeably called probability distributions and PDFs.

The probability density function (PDF) is used to specify the probability of the random variable *equaling* a value in the sample space, whereas the cumulative distribution function (CDF) is the probability that the variable takes a value *less than* *or equal* to a value in the sample space. The PDF and the CDF are non-negative everywhere and the integral over the entire space equals 1.

The finite differences are important but are also one reason why even savvy professionals shy away from Monte Carlo simulations. To keep it simple, we will refer to everything as a probability distribution in this discussion.

**Three Probability Approaches with Pros and Cons**

**1. Good Lifetime Data with Classic Statistics **

This is the standard statistical approach. On the one hand, it is nirvana (an ideal or idyllic place); on the other hand, it is frequently like the lonely wanderer in search of El Dorado (a mythical place of fabulous wealth and opportunity).

**Pros**

Most definitive

This is what we are all hoping to have and use

**Cons**

Laboratory testing does not reflect actual operating conditions

Field testing is expensive and often not practical

Field data inherently inconsistent due to updated equipment or changed contexts

Running controlled tests to failure takes time and is expensive

Data capture from work orders is erratic and of questionable quality

In practice, we seldom run important things to failure

Few organizations do formal Root Cause Failure Analysis (RCFA)

**2. Distribution Fitting & Bootstrapping**

This approach applies to partial lifetime data. Efron and Tibshirani (1993) are credited with the computational approach of the bootstrapping method.

__Pros__

Proven and accepted approach

Distribution fitting is a standard tool in most Monte Carlo simulation software

__Cons__

Frequently develop distribution curves beyond the range of available data

Requires some knowledge of the typical distribution for the class/type of data

**3. Build Your Own Probability Distributions (Cumulative Density Function)**

This approach is applicable when there is limited data or no data. It is an Interview-based approach with roots in Howard (1960s) and specific methods like probability wheels developed by Spetzler and Holstein (1974).

__Pros__

Proven and accepted approach

A common approach in practice. Either stand-alone or complimenting bootstrapping.

__Cons__

Requires some knowledge of the typical distribution for the class/types of data

Reliance on subject matter experts

Highly dependent on qualitative survey skills (facilitation, administration, and analysis)

Proprietary tools claiming to produce better results than others

**Get In the Game**

The world is richly skewed and highly uncertain. Use Monte Carlo simulations in your funding forecasts for future infrastructure and facility needs. It helps to have a trusted partner helping you build your forecast model, but you do not need to be able to recall all of your college statistics or be an expert at data collection to do it.

There are many resources to help you. The first step is to get in the game.

__JD Solomon Inc__* provides program development, asset management, and facilitation services at the nexus of facilities, infrastructure, and the natural environment. Monte Carlo simulations are a standard tool in our approach to addressing risk and uncertainty. Contact us for more information on developing lifecycle forecasts, assessing the total cost of ownership, or providing third-party reviews of capital improvement programs or operating budgets.*