Statistics mean duration rate

In principle, FCS can also measure very slow processes. In this limit the measurements are constrained by the stability of the system and the patience of the investigator. Because FCS requires the statistical analysis of many fluctuations to yield an accurate estimation of rate parameters, the slower the typical fluctuation, the longer the time required for the measurement. The fractional error of an FCS measurement, expressed as the root mean square of fluorescence fluctuations divided by the mean fluorescence, varies as 1V-1/2, where N is the number of fluctuations that are measured. If the characteristic lifetime of a fluctuation is r, the duration of a measurement to achieve a fractional error of E = N l,/- is T = Nr. Suppose, for example, that r = 1 s. If 1% accuracy is desired, N = 104 and so T = 104 s. 

In the meantime, a reasonably fast radioactive decay makes a different technique for obtaining data on fjp feasible, even with rather low activities. One can measure the fraction r]c of the nuclei introduced into the IC column which survives at its exit. This principle can be used in on-line experiments with the nuclides which have mean lifetimes much less than the nominal duration of the run in practice, it means the range from seconds to hours. For a nuclide with the particular tx, two or more measurements at different temperatures must be done. At least one at a temperature when t -C tx to find the production rate of the detectable activity, as well as one when r c is of the order of lx and so the surviving fraction is in the range 0 < rf < 1. From the point of view of the statistics, most desirable is to aim at rjc near 0.5. Obviously, r = exp -t r/tx), hence ... 

To illustrate, consider a uniform cash flow of 1000 per year beginning at some uncertain time m and continuing for a duration of t years. The delay to initiation is uniformly distributed between 6 months and 1 year. The project duration is gamma distributed with mean of 3 years and standard deviation of 1 year the parameters of the gamma distribution yielding these statistics are a = 3 and b = 9. The nominal interest rate is 10% compounded continuously. It is assumed that the initiation time and project dmation are independent random variables. Our problem is to determine the equivalent present value of these cash flows. (This problem is taken from Park and Sharp-Bette (1990, p. 411)... 

The mean squared energies (A ( o)) are of course also determined by the intermolecular potentials. The duration of the collision or the lifetime of the collision complex will be of primary importance. The statistical collision model assumes a statistical distribution of the energies of all oscillators in A and M during collision. If before collision A is highly excited but M is not excited, this results in very effective energy transfer. With the statistical theory of reaction rates as discussed in section 1.8 one can easily calculate for this model values of (AE ( o)>. see e.g. ref. 97. One finds in general V kT, and so = 1 in equation (1.55). Details of (AE (Eg)) for this model are... 

First, a fault tree of the standby power plant is established. This fault tree should be quantitatively evaluated by means of the failure rates, the duration of operation, inspection intervals, and repair times. The result will be a statistically safe statement of the failure probability of the standby power plant. 

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