# Probability Distributions Intro

March 29, 2018

Back to: Random Testing

When a more detailed description of a random variable is desired, beyond the point estimates covered in the Statistics lessons above, a *probability density function* (PDF) can be used. A PDF is a histogram of the probability of the different levels of the variable. When a signal is digitized into a series of values, *x _{n}*,

*n*= 1, …,

*N*, they are considered a set of

*samples*of the signal. The probability histogram is computed by counting the number of samples,

*m*, with values fitting within a set of

*bins*covering the entire range of the signal amplitude. This is illustrated in Fig. 4 using the car vibration data from Fig. 1. The fraction of samples in each bin,

*m*/

*N*, is plotted vs. the bin level as a bar chart (with the bin width being

*Δ*

*x*= 0.02 G in this case). In this form the sum of all the bin values equals 1, but their range of amplitudes depends on the value of

*Δ*

*x*.

*Figure 4. Histogram of the Car Vibration Signal shown in Fig. 1*

The probability *density* is computed by dividing the fraction of samples in each bin by the bin width, *Δ**x* : PDF = *m* / (*N* *Δ**x*). This is plotted in Fig. 5a as a line graph. The PDF can be normalized further by plotting it vs. the number of standard deviations from the mean, *z* = (*x* – *μ*) /*σ* , as shown in Fig. 5b. These are called *sample* PDF’s. The true PDF is found in the limit as *N* → ∞ and the number of bins → ∞ (with the bin sizes becoming infinitesimal). In either case the area under the PDF curve is 1.

*Figure 5. Sample PDF’s of the Vibration Signal shown in Fig. 1.*

The following lessons examine the PDFs for some ideal statistical distributions. For these distributions the PDF can be described as a continuous function of the amplitude, *x*, and plays a significant role understanding random vibration analysis.