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Probability Distributions

Our statistical analyses are based on the concept that if we measure a sample multiple times, then the results would represent some known distribution. For most of the measurements we make in safeguards, this is a Gaussian (or Normal) Distribution. The Gaussian distribution is given by:

gaussian

where μ is the mean for the distribution, σ is the standard deviation of the distribution, and x is any measured value. This distribution can be used to determine the probability that any additional measurement will fall within some interval from x1 to x2, or for shorthand we say that it will be in the interval (x1,x2). Thus we refer to p(x) as a probability distribution function (PDF). The probability that a measurement will fall within the interval dx is given by p(x)dx. So the probability that a measurement will fall in the interval (x1,x2) is given by

prob_equ_1

Since the measured value must fall somewhere between -∞ and +∞, we can say that

prob_equ_2

or that the probability that the next measurement will fall between -∞ and +∞ has to be equal to unity (i.e., it is absolutely certain that it will).

It is convenient to define the cumulative PDF given by

prob_equ_3

This function is equal to 0 when x=-∞, and it is equal to 1 when x =+∞. A plot of p(x) and P(x) for a Gaussian distribution is shown in Figure 7.

image14

Figure 7. Probability distribution function [p(x)] and cumulative probability distribution function [P(x)] for a Gaussian distribution (for a mean of 0 and a standard deviation of 1).

The distribution of measured values considered in the previous section could be well approximated by a Gaussian distribution. Figure 8 shows the frequency plot of measured values (for the higher uncertainty set of 155 measured values) and the Gaussian distribution on the same plot. As can be seen, they match up rather well. The Gaussian distribution was normalized such that the integral of the distribution over the whole domain is equal to the total number of measurements performed.

image15t

Figure 8. Frequency of measured values and Gaussian distribution for the 155 measurements from the previous section.

Using a Gaussian distribution to represent the data allows us to make a number of useful conclusions about the measurement system. The distribution is such that if we make another measurement, we expect that there is a 68% chance that it will fall in the interval (μ-σ,μ+σ). There is a 95% chance that it will fall in the interval (μ-2σ,μ+2σ) and there is a 99.7% chance that it will fall in the interval (μ-3σ,μ+3σ).

 

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