asurement apparatus. Probability distributions Next, we consider the continuous-exponential distribution. The PDF is defined for x > O as follows: p(x) = le-ir (2.17) where 2 is a strictly positive constant called the rate parameter. The mean and variance of the exponential PDF are <x> = 1/2 and o2 = 1/22, respectively (this is left as an exercise). Between the discrete-exponential and the continuous-exponential PDFs, there is no one-to-one correspondence, except in the limit of large means. Indeed, setting N = 1/2 in Eq. (2.17), we can redefine the discrete-exponential PDF (x = n integer) under the (2.18) Such a definition corresponds to that of the continuous PDF (Eq. 2.17) only in the limit 0 (for which log( + 2) -> 2), which corresponds to the limit N The continuous-exponential PDF is used to model the timing of physical events that happen at a constant average rate. If x = t is a time variable, the event occurs at exact time t with probability p(t) = 2 exp(-)t). We note that in this case, 2 has the dimension of an inverse time (e.g., inverse seconds or inverse days), meaning that the PDF is a probability rate (in /s or /day units) rather than being dimensionless. The exponential PDF is a maximum for t = 0 and decays rapidly as time increases. On average, the events occur at time <t> = 1/2 = T, where T is a characteristic time constant. In atomic physics, T is called the decay constant, or also the 1/e lifetime. For radioactive atoms or for atoms used in laser materials, this means that the disintegration or photon emission occurs in average at the time t = T. The probability that any atom decays at a time after t is given by (2.19) = exp(-t/t). The integration expresses the fact that the probability p(T > t) is given by the continuous sum of all probabilities p(t') where t' belongs to the time interval [t, +00]. The result shows that the probability that any atom decays after t = T is (T > T) = e-superscript(1) = 1/e 0.36, hence the name 1/e lifetime. The probabilities that atoms decay after times t = 2t, 3t