Exponential distribution, mean lifetime

A typical dipole moves in a potential well without friction during the lifetime tv between two consecutive strong collisions. The well, the form of which should preferably be taken as simple as possible, is chosen on the basis of a phenomenological approach based on the experience gained in previous studies (VIG, GT). The time tv is interpreted as a duration of local order in a liquid. The exponential distribution over fv is used, where the mean lifetime r (which later is referred to simply as lifetime ), r = (fv), is a free parameter of the model. 

When members arrive according to a Poisson process with rate A and the time that each active member stays in the group (which is referred as lifetime) is exponentially distributed with mean 1 / p, the expectation of S can be computed as follows ... 

What is the meaning of these peaked dwell time distributions Mathematically, the dwell time distribution is the convolution of the individual exponential distributions governing the underling state lifetimes [79]. If one of these lifetimes is much longer than the others, we would expect to see a distribution dominated by a single exponential decay. However, if multiple kinetic states have lifetimes of comparable duration, these states can be thought of... 

Recordings from muscle-type nicotinic receptors contain many brief closures [see Figs, ll.lland 11.13 (below)],with a mean lifetime of around 15 ps at 20 C, and because the shortest event that can be detected reliably is around 20-30 ps, the majority of these are missed (because the durations are exponentially distributed, it is possible to estimate the mean even when observations as short as the mean are missed). Methods have improved since then. Now an exact method for allowing for missed events is available, so it is possible to analyze an entire observed recording by maximum likelihood methods that extract all of the information and that incorporate missed event correction. There are some other methods under development, particularly methods based on the theory of Hidden Markov processes, but none are in routine use. 

This study shows that none of the various forms of relaxation function used to describe ageing are completely satisfactoiy and TRS is inappropriate. Correlation between results, (Figure 9) indicates the inherent connectivity between the processes. Curro et al (24,25) have studied the change in density fluctuation with temperature and annealing time for PMMA (25) and compared it with specific volume data. Positron annihilation data on PMMA (27,28) has been interpreted in terms of free volume. For a distribution of hole sizes there will exist many decaying exponentials each with a different characteristic lifetime. The composite of these many exponentials can itself be approximated to an exponential, and it is this decay constant that is used to represent the mean lifetime, and therefore mean hole size. 

Consider a life testing experiment where n items are simultaneously put on test at the outset and are not replaced on failure. Let Xi denote the lifetime of component i,i = n. Suppose that Y,- follows an exponential distribution with common mean lifetime 9. That is, Xt has a probability density f(x 9) =, x > 0,9 > 0. 

The density function of the logarithm of exponentially distributed lifetimes. Using this trick, the estimation of the mean life can be reduced to the determination of the mode (i.e., the most frequent value) of a logarithmized lifetime histogram related to the above distribution. This method can be used whenever the number of lifetimes that can be measured is so small that the standard method of mean life determination (r = 1/A) based on the exponential fit of lifetime data fails. See also remark ( 35)... 

In Sect. 2.8 we saw that the mean lifetime r, of a molecular level E/, which decays exponentially by spontaneous emission, is related to the Einstein coefficient Ai by Ti = 1/A/. Replacing the classical damping constant y by the spontaneous transition probability A/, we can use the classical formulas (3.9-3.11) as a correct description of the frequency distribution of spontaneous emission and its linewidth. The natural halfwidth of a spectral line spontaneously emitted from the level Ei is, according to (3.11),... 

In calculation of capacitor reliability (Lindquist, 1977) it is assumed that capacitors have lifetimes that are exponentially distributed. In this case the distribution is characterized by one parameter only, namely, the mean time to failure (MTTF) tq. The inverse of the MTTF is the failure rate of the component. 

Generate a random number according to exponential distribution the mean of which is MTBF as a lifetime of the unit. 

Supposing that a source array of sonar contains 1000 components, and 10% component failures are allowable. The lifetime of each component follows a same exponential distribution, the mean of which is 20,000 hours. The source array works 24 hours a day without closing down. MTBCF of the source array is demanded to be 35,000 hours in the contract. If maintenance personnel repair the source array every three months FDR, FIR and RR are all supposed to be 0.98, does the source array satisfy its demand on MTBCF ... 

Using the cell-attached patch clamp technique on frog muscle fibers (79), one can observe only two conditions the open, conducting state of the receptor and a nonconducting state of unknown identity. The transitions behave according to stochastic principles the lifetimes of any particular condition are distributed exponentially. The open state has a mean duration that is the inverse of the rate of channel closing. Because channel open time depends only upon a conformational shift, agonist concentration does not influence the parameter. It is, however, influenced... 

In this part we present an example for the pressure dependence of hole-size distribution. In our experiments we could not vary the temperature but keep it constant at room temperature. Details of experiments are described by Goworek [2007]. Figure 11.7a displays the mean, T3 (= (T3)), and standard deviation, <73, of o-Ps lifetime distribution and the o-Ps intensity h of PIB at 296 K as a function of pressure, P [Kilburn et al., 2006]. All of these parameters exhibit an exponential-like decrease with increasing pressure, h shows a small hysteresis which can be attributed to positron irradiation effects. T3 decreases from 2.0 ns at P = 0.1 MPa to 0.83 ns at 1.3 GPa and from 0.45 ns to about 0.05 ns. The low value of T3 =0.83 ns is possibly the lowest o-Ps lifetime observed until now for polymers. A lifetime of 0.5 ns is the theoretical limit of the pickoff annihilation for disappearing hole sizes [see Eq. (11.3)]. In polytetratluoroethylene (PTFE) a second, medium o-Ps lifetime of about 1 ns has been resolved and attributed to o-Ps annihilation in the densely packed polymer crystals [Dlubek et al., 2005d]. 

A more common method for medical devices is to run the life test until failure occurs. Then an exponential model can be used to calculate the percentage survivability. Using a chi-square distribution, limits of confidence on this calculation can be established. These calculations assume that a failure is equally likely to occur at any time. If this assumption is unreasonable (e.g., if there are a number of early failures), it may be necessary to use a Weibull model to calculate the mean time to failure. This statistical model requires the determination of two parameters and is much more difficult to apply to a test that some devices survived. In the heart-valve industry, lifetime prediction based on S-N (stress versus number of cycles) or damage-tolerant approaches is required. These methods require fatigue testing and ability to predict crack growth. " ... 

For R6G on silica, experiments were performed on populations of molecules to obtain the mean, unperturbed lifetime, Zf, and its distribution. Five unperturbed fluorescence decay curves were measured for tips positioned 1.0 to 1.1 pm above high-coverage surfaces ( 10 to 10 molecules illuminated) (Fig. 11(D)). The decay curves were fitted to a single exponential with Zf = 3.65 + 0.04 ns, which is in good agreement with the 3.5+ 0.1 ns obtained previously [22]. Statistical noise and instrument nonlinearity place an upper limit on a possible Gaussian standard devia-... 

In the case of the lifetime distribution of radionuclides, excited states, etc., the expected value t is called the mean life, while the median Ti,2 is referred to as the half-life. The explanation for the name half-life is given in the next subsection on the exponential law of radioactive decay. It will be shown that the half-life is independent of the time elapsed, which is an obvious proof of the agelessness of radionuclides. Note that physicists often use the term lifetime not only in the sense it is used in this chapter, but also in the sense mean life. Fortunately, in the really important cases, i.e., when quantitative statements are made (e.g., the lifetime of the radionuclide is 10 s ), the ambiguity is removed and the reader can be sure that such a statement actually refers to the mean life. 

Equ.(6) is a special case of the more general statement, that the lifetime distribution of a given state v is (in simple cases) an exponential with a mean ... 

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