Common Formulas

Running through a couple of practice CRE exams recently (yeah, I know I should get out more…) found a few formulas kept coming up in the questions. While it is not a complete list of equation you’ll need for the exam, the following five will help in many of the questions. They seem popular maybe because the relate to key concepts in the body of knowledge, or they are easy to use in question creation. I do not know why.

The exponential reliability function.

$$ \large\displaystyle R\left( t \right)={{e}^{-\lambda t}}={{e}^{-{}^{t}\!\!\diagup\!\!{}_{\theta }\;}}\text{, where }\theta ={}^{1}\!\!\diagup\!\!{}_{\lambda }\;$$

This formula provides the probably of success at time t given either the failure rate, λ, or the MTBF (or MTTF), θ.

Note: as many of you know, I do not like the use of MTBF in general and would prefer the exponential distribution to find less prominence in the CRE Body of Knowledge, yet it is there and probably the most common formula used in the exam. Alas. Once you get your certification or want to improve your reliability engineering skills, see my other blog at nomtbf.com.

The failure  rate, λ, or the MTBF (or MTTF), θ, are determined using the simple formula

$$ \large\displaystyle \theta ={}^{1}\!\!\diagup\!\!{}_{\lambda }\;=\frac{\text{Total time}}{\text{number of failures}}$$

The total time is all time the units are on test. So, if there are three units tested for 500 hours and one fails at 400 hours (not replaced), the total time is 500 + 500 + 400 = 1,400 hours. And the total number of failures is one. Thus we would find θ = 1,400 / 1 = 1,400 hours. The inverse is the failure rate.

See the post on Exponential Reliability for more details.

The next two are related as they deal with reliability modeling using reliability block diagrams, RBD. The series model has units arranged such that any one item that fails causes the system to fail. The formula for three units is

$$ \large\displaystyle {{R}_{system}}={{R}_{1}}\times {{R}_{2}}\times {{R}_{3}}$$

This obviously generalizes to any string of items in series. A nice trick is when all the individual items are described by an exponential distribution, one then can add the failure rates (not MTBF’s) to find the system failure rate, then do the exponential calculation once to find the system reliability.

The related formula is for the parallel structure. If two items are in parallel, then the formula is

$$ \large\displaystyle {{R}_{system}}=1-\left[ \left( 1-{{R}_{1}} \right)\left( 1-{{R}_{2}} \right) \right]$$

The 1-R is the unreliability at time t, which permits multiplying the unreliabilities as they are now in a series structure, then another 1 minus the result to bring back to reliability. this again is scalable for any number of units in parallel.

See this list of posts for more details around these concepts and formulas.

The last formula is the binomial.

$$ \large\displaystyle P\left( x,n,p \right)=\left( \begin{array}{l}n\\x\end{array} \right){{p}^{x}}{{\left( 1-p \right)}^{n-x}}$$

Only useful when an experiment only has two possible outcomes (i.e. pass/fail, blue/green, etc.) The formal above is the probability of exactly x successes in n trials with a probability of success equal to p on each trial.

Looks like I need to write an article on the binomial distribution.

Each of these formulas appeared a few times in each practice exam I did. Of course, your exam may be quite different, yet knowing these formulas and how to use them will serve you well as a reliability professional.

What do you see as the most common formulas? Let me know if I need to add to the above list.

Related:

Exponential Reliability (article)

The Exponential Distribution (article)

Using The Exponential Distribution Reliability Function (article)