Counting distributions

In principle, the statistics of radioactive decay are binomial in nature. If we were to toss a handful of coins onto a table and then examine the arrangement, we would find coins in one of two dispositions - heads up or tails up. Similarly, if we could prepare a radioactive source and, during a particular period of time, monitor each individual 

Let us suppose that we could determine exactly which of the atoms, and how many, decayed during the count period. If we were able to repeat the experiment, we would find that different atoms and a different number of atoms decayed in the same period of time. We can regard each such measurement, each count, as a sample in the statistical sense, an attempt to estimate the true decay rate. We would expect the distribution of these counts to fit a binomial distribution (sometimes called a Bernoulli distribution). This distribution applies because  

If we consider each atom in our source there is a certain probability, p, that the atom will decay during the period we choose to make our measurement. This probability is related to the decay constant of the atom and it is straightforward to demonstrate that  

Regardless of the shape of the distribution, the most likely number of decays is given by Equation (5.19)  

Taking the square root of the variance, we can calculate the standard deviation and, for the three specific cases, this would be 1.34, 2.24 and 1.34 decays (or counts, assuming 100% efficiency). Equation (5.19) is interesting in that it predicts that as the probability becomes very small or very near to 1, the width of the distribution or, we might say, the uncertainty on the number of decays, tends to zero. This is not unreasonable. If / = 1, we can expect all atoms to decay and if = 0 none to decay. In either case there is no uncertainty about the number of decays which would be observed. 

---

Roche datasets, (b) Cumulative topological polar surface area (A ) distributions of compounds in the Gasteiger [33] and Roche datasets, (c) Cumulative chemical complexity distributions of compounds in the Gasteiger [33] and Roche datasets, (d) Cumulative rotatable bond count distributions of compounds in the Gasteiger [33] and Roche datasets. 

Low-Pressure Impactor Data. Table I lists the count medians and geometric standard deviations from log-normal fits of the distributions obtained by SEM analyses of sample MKV-1, Listed also are the corresponding mass medians calculated from the count distributions. 

Fourier transform -Autocorrelation function -Moments of photon count distribution... 

The calculation of a PDF begins with experimental data in the form of a photon count distribution F(j), defined in Table 1 along with the other functions discussed here. 

The factorial moments derived from the count distribution are equal to the zero moments, Zm, of the PDF, and simply related to the moments about its average value, Cm- (Table 1). The moments alone provide significant information about the concen-... 

Table 1. Basic Quantities in Analyses of CW Laser Scattering for Probability Density Function. In Eq. 1 within the table, F(J) is the photon count distribution obtained over a large number of consecutive short periods. For example, F(3) expresses the fraction of periods during which three photons are detected. The PDF, P(x), characterizes the statistical behavior of a fluctuating concentration. Eq. 1 describes the relationship between Fj and P(x) provided that the effects of dead time and detector imperfections such as multiple pulsing can be neglected. In order to simplify notation, the concentration is expressed in terms of the equivalent average number of counts per period, x. The normalized factorial moments and zero moments of the PDF can be shown to be equal by substitution of Eq.l into Eq.2. The relationship between central and zero moments is established by expansion of (x-a)m in Eq.(4). The trial PDF [Eq.(5)] is composed of a sum of k discrete concentration components of amplitude Ak at density xk. [The functions 5 (x-xk) are delta functions.]...

Relationship between photon count distribution and concentration PDF... 

In Fig. 2 we show a quasi-sinusoidal LED output as measured from the unattenuated signal on the oscilloscope, and the pulse count distributions obtained with ND1 and ND3 filters (average counts per 400 ysec channel of 100 and 1, respectively). Also shown in this figure are two typical PDF fits to the latter data (average count per cycle = 1), compared to the PDF calculated from the LED signal variation displayed on the oscilloscope. 

The figure illustrates that the 100 count per period count distribution reproduces the shape of the PDF fairly well, whereas the 1 count per period distribution displays no obvious resemblance to the PDF. However, the typical fits calculated from two independent sets of 1 count per period data, shown in Figs. (2D) and (2E), reproduce the PDF to an accuracy which approaches that of the 100 count per period result. The unoptimized Fortran program which produced these fits requires approximately 20 seconds per rtjn on a Honeywell DPS-2 computer. Our experience with this and other shapes for the PDF leads to a conclusion that the overall shape of a widely distributed PDF can be obtained reliably, though some ambiguity is found in the finer details at average counts as low as one per period. As the average count is increased to two or four per period, the resolution improves steadily. 

Figure 1. Experimental configuration to obtain detected photon-count distributions using a LED source...

Fig. 12.6. Grain count distribution. L cells labelled for 10 min with [3H]thymidine (2.5/1 Ci/ml 0.36 Ci/mmol) and processed for autoradiography using NTB3 emulsion. Those cells (62% of the total) with one or more grains are recorded. A Poisson distribution with a mean of 30 is included for comparison. (Reproduced from Cleaver, 1967, with kind permission of the author.)...

The center of a glass slide has 100 ruled squares. Over the area occupied by these squares was placed a small amount of xylene suspension containing particles. After drying the distribution over the rulings was examined. The following count distribution was obtained ... 

Figure 4.4 shows a typical frequency count distribution of Ucnt obtained by this procedure with a 75 gm cylindrical active electrode in 20 wt% NaOH electrolyte. The measurements were repeated 200 times. It can be seen that the critical voltage is widely distributed and follows a Gaussian distribution. The first moment of this distribution defines the mean critical voltage (E/" ). 

It has already been stated that erg is identical for plots of the distribution of any parameter related to particle diameter, e.g. mass or surface area. Log-probability plots of count distribution,... 

Histologic features in the airways of patients with OA are similar to those found in individuals with non-occupational asthma (NOA). Common pathologic sequelae include airway smooth muscle hypertrophy, mucosal oedema, increased mucus production from greater numbers of mucous secreting glands and goblet cells, and deposition of collagen beneath the basement membrane (Fabbri et al., 1993). BAL fluids from normal, NOA and OA subjects have similar inflammatory-cell count distributions (90 to 95 per cent alveolar... 

It is often necessary to determine the statistical error involved in this calculation, following the procedures given in Section 19.5. If the background count distributions around Ci and Ca appear to be horizontal, it is often advisable to take averages of several ehannels preceding the FEP and several channels following the FEP in order to obtain a more representative baseline correction. There are several more so-phistieated methods of baseline eorrection, but their discussion is beyond the scope of this ehapter. 

First, we present a simple algorithm for generating random fluorescence based on the theory presented in Section II, by using the stochastic Bloch equation, Eq. (4.6) and the classical photon counting distribution, Eq. (4.18). A measurement of the spectral trail is performed from t = 0 to t = ie d. As in the experimental situation, we divide into N time bins each of which has a length of time T For each bin time T, a random number of photon counts is recorded. Simulations are performed following the steps described below ... 

In Figure 4.2, we show the evolution of the photon counting distribution Pk(n), Eq. (4.18), at a fixed laser frequency, chosen here as —c o = —V = — 5F. The spectral diffusion process is identical to the one shown in Figure 4.1. Here, k denotes the measurement performed during the kth time bin as described in Step 3. Notice that two distinct forms of the photon counting distributions appear. During the dark period at the chosen frequency — Oq = Figure 4.1 (e.g., 2.1 x 10" < Ft < 3.5 x lO ... 

Figure 4.4. Time evolution of photon counting distributions P in) for the fast modulation case at = (Og in Figure 4.3 is shown. Other parameters are given in Figure 4.3.

The first tests on the X-ray camera were performed in order to verify the effective size of the pixel, that is the area where the events are detected contemporarely by two orthogonal microstrips. In order to do this the detector was scatmed by a collimated beam in both directions and for each position the number of the coincidence events from two orthogonal electrodes was recorded. In Fig. 2 are shown the count distributions along one direction for pixels (1,2) and (1,4). Because the diameter of the source spot is much larger than the width of the pixels the two distributions are overlapped... 

An additional source of error to n is introduced by the truncation of the experimental count distribution P(n At) by the digital nature of the experiments. Thus, for m such that TMP(m At ) <1, the experimental estimator n given by... 

Figure 2.3 PCH for an ideal scatterer placed at the laser focus. The experimental photon-count distribution (solid circles) is exactly fit by a Poisson function (solid line) with an average number of photon counts = 6.7. Residuals are shown in units of standard deviations. The fit gives = 0.91. A total of 131072...

In a similar way to which PCH analysis is applied to single molecule fluorescence data sets (see Section 2.3), analysis of the first three moments of the photon count distribution can be used to determine certain properties of a sample, for example the concentrations of fluorescent species [11]. However, moment analysis is rather rare in single molecule spectroscopy and has effectively been superseded by methods that use a complete description of the photon count distribution such as PCH and FID A. 

Higher order autocorrelation analysis is analogous to the analysis of the higher order moments of the photon count distribution. Such a correlation function is given by. 

Figure 3.17 Photon count histogram of an ideal scatterer placed at the laser focus. The collected photon count distribution (circles, normalized) is fitted exactly by a Poissonian function (line) indicating that there are no fluctuations in the detected signal arising from instability of the light source or other instrumentation.

In practice, we do not know ag, the standard deviation (or uncertainty) of the net background count distribution. All we do have are the sample and background estimates. Taking Equation (5.39) again and remembering that var(count) = count, we can deduce that ... 

If the number of counts accumulated is small, then, even though the count distribution will be Poisson, the approximation to a Normal distribution will not be valid. This means that the relationships to calculate the decision limits given above will not be valid. For number of counts of less than 25, we must resort to the Poisson distribution itself. 

---