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Probability Distributions Intro

March 29, 2018

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, xnn = 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 Δx = 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 Δx.

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

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, Δx :  PDF = m / (N Δx).  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.

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