# Statistics & Probabilities Introduction

March 29, 2018

Back to: Random Testing

At first, it may seem difficult to describe a random vibration signal quantitatively. Random waveforms are inherently irregular, non-deterministic, and non-repetitive. We cannot use a measured point on a random time-history graph to predict any point before or after. However, we can analyze a random waveform from a statistical standpoint because of the assumption that random vibration is stationary.

#### Stationary Process

Random vibration is a stationary process, meaning a random waveform’s characteristics are statistically similar at any point in time. If we measure the vibration of a metal beam today and next year, the root-mean-square (RMS) value will not differ significantly (given that its structure has not changed). Since we can assume random vibration is stationary, we can run the same test on equivalent components and define test standards for specific devices/structures.

### Random Statistics & Probabilities

This section covers some of the ways to quantify random signals, including statistical measures such as the mean value and probability distributions such as the bell curve. It also covers product design applications such as tolerances and fatigue analysis.

In this section, we assume that a continuous vibration signal, *x*(*t*), has been digitized into a sequence of numbers, *x _{n}*,

*n*= 1, …,

*N*, for analysis.

#### Main Takeaways

- How to analyze random vibration using statistics
- Control parameters related to random vibration testing

For an overview of the topics presented in this course, consider reading the following paper: Statistical Properties of the Random PSD.