re 82%, but this prediction covers an infinite amount of time. It is possible to make a more detailed failure prediction for a given period spanning times t1 to t2 > t1. Indeed, the probability of getting a failure between these two times is: <T<t2)=1- - [p(T < t1) + p(T > t2)] = - [1 - exp(-2t1) + exp(-2t2)] (2.20) = exp(-2t1) - exp(-tt2). With the above formula, one can determine the failure probabilities concerning any specific periods defined by [t1, t2]. Next, we consider as a last but key example, another continuous PDF, which is the Gaussian or normal distribution. With a mean <x> = N and a variance o2, it is formally defined according to: (2.21) Since the function exp(-u2) is symmetrical with a peak value centered at = 0, we see that the Gaussian PDF is centered about its mean, <x> = N, with a peak value of p(N) = P peak = 1/(0V2). For values = N O T 2, we observe from the definition that the probability drops to e-1 P peak = Figure 2.6 shows plots of Gaussian PDFs with mean N = 0 and different standard deviations. The characteristic bell shape has justified over time the popular name of bell distri- bution, which is well known to a large public. The surface S under the curve, which is defined by two points X1, X2, i.e., (2.22) represents the probability of event x taking a value in the interval [x1,x2]. It can be shown by integration in Eq. (2.22) that p(N 0.682, meaning that Probability distributions <x>=N=0 0 = 0.5 Figure 2.6 Gaussian probability distribution with mean <x> = N = 0 and standard deviations o = 0.5, 0.75, 1, and 2. 68.27% of the bell surface concerns events falling within two standard deviations (to) of the mean (N). Likewise, for the intervals 20 and 30 about the mean, the surfaces represent 95.4% and 99.7 of the total bell surface, respectively. A physical parameter obeying a Gaussian PDF, for instance, electrical noise in radio or TV signals, is experimentally characterized through discrete sampling, even if the measurement apparatus (e.g., an analog oscilloscope) provides a continuous