Exponential probability density

HANNA, S.R. The exponential probability density function and concentration fluctuations in smoke plumes, Boundary Layer Met., 29, 361-375, 1984. 

Both the RBC distribution (8) and the geometric distribution (11) are defined only for specific integer bubble sizes, and derivatives of their distribution functions do not exist. For subsequent developments we need an equivalent continuous distribution. Fortunately, for N and k large with respect to m, both discrete distributions can be closely approximated by the exponential distribution if its mean is set to the RBC mean volume given by (10). The exponential probability density is... 

The functions exppdf(x,mu) and expcdf(x,mu) are available in the MATLAB Statistics Toolbox for calculating the exponential probability density and exponential cumulative distribution function, respectively. 

The lifetime in years of liquid crystal display (LCD) is a random variable with the exponential probability density function given by ... 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

Probably, a similar procedure was previously used (see Refs. 1 and 93-95) for summation of the set of moments of the first passage time, when exponential distribution of the first passage time probability density was demonstrated for the case of a high potential barrier in comparison with noise intensity. 

It is noted that the solutions are real rather than complex exponentials. Since the constant D = 0 the probability density remains finite as x —> oo. In the region x < 0, as before,... 

In a first simple model for target detection it is assumed that the background clutter can be described by a statistical model in which the different range cells inside the sliding window contain statistically independent identically exponentially distributed (iid) random variables X i,..., X/v. The probability density function (pdf) of exponentially distributed clutter variables is fully described by the equation ... 

The temporal luminescence of a highly heterogeneous sensor-carrier mixtures cannot be uniquely represented by sums of exponentials (Eq. (9.23)) due to the lack of orthogonality of the exponential function. In this case it becomes appropriate to express equations (9.17) or (9.23) in terms of probability density functions or lifetime distribution functions 5t(8 14)... 

Figure 1.2 The probability density as a function of time in atomic units. The probability at time zero is given by p(x, 0) = 2 The decay in the interaction region can be well approximated as exponential.

Figure 1.7 The probability density of the wave packet at time t = 200 on a logarithmic scale. Note that outside the interaction region, we have an exponentially diverging function.

Outside the interaction region, the probability density seems to increase exponentially in space. 

Conversely, a coherent superposition of continuum states with a population closely reproducing an isolated peak in the density of states, which corresponds to a resonance, can be built in such a way to give rise to a localized state. From this localized state, there will be an outward probability density flux, i.e., it will have a finite lifetime. In the limit of a resonance position far from any ionization threshold and a narrow energy width, the decay rate will be exponential with the rate constant T/ft. The decay is to all the available open channels, in proportion to their partial widths. 

It is noted that the right-hand side of Eq. (10.20) is just the series expansion of an exponential function. Therefore the overall residence time distribution probability density function in the SCISR is obtained to be... 

One has simply to assume a particular probability distribution for A with the survival function available in a closed form, namely the exponential, Erlang, Rayleigh, and Weibull. Table 9.1 summarizes the probability density functions, survival functions, and hazard rates for the above-mentioned distributions. In these expressions, A is the scale parameter and p and v are shape parameters with k, A, p > 0 and v = 1, 2,.... ... 

Actually, the inverse problem should be solved, i.e., given the data n(t) containing errors, obtain a plausible candidate / (h) associated with a known function p(t,h). This function, termed kernel, is assumed to be a retentiontime distribution other than an exponential one otherwise, the problem has a tractable solution by means of the moment generating functions as presented earlier. This part aims to supply some indications on how to select the density of h. For a given probability density function f (h), one has to mix the kernel with / (h) ... 

The physical meaning of the strange variable in the Hermit polynomials is that the distance of the nodes of these functions keeps getting smaller with the progression of time by the exponential law e °jt. According to this result, the probability density of the nth energy state with displacement q is... 

Notice that in the Continuous-Time Random Walk (CTRW) as used in Klafter et al. [50], in the case where the waiting time distribution is exponential, i(t) = a expf at], the same evolution for the probability density p(x,t) and the phase-space distribution ct(x, t) occurs as that resulting from Eq. [57], This can... 

---