Poisson distribution standard deviation

To determine the number of samples required to achieve an experimental standard deviation (S) equal to or less than the Poisson distribution standard deviation (a), a 1 cc/sec gas flow rate and four minute samples were used. The particles were generated with a PUS, Inc. particle generator model PG-100, providing a particle count of 2000 partlcles/ft or greater. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

The gaussian distribution is a good example of a case where the mean and standard derivation are good measures of the center of the distribution and its spread about the center . This is indicated by an inspection of Fig. 3-3, which shows that the mean gives the location of the central peak of the density, and the standard deviation is the distance from the mean where the density has fallen to e 112 = 0.607 its peak value. Another indication that the standard deviation is a good measure of spread in this case is that 68% of the probability under the density function is located within one standard deviation of the mean. A similar discussion can be given for the Poisson distribution. The details are left as an exercise. 

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) ... 

The discrete Poisson distribution is only characterized by one parameter, the mean Y. The standard deviation is given by sY = Jy and the relative standard deviation by SyreI = 1 / JY. 

Poisson-distributed noise, however, has an interesting characteristic for Poisson-distributed noise, the expected standard deviation of the data is equal to the square root of the expected mean of the data ([11], p. 714), and therefore the variance of the data is equal (and note, that is equal, not merely proportional) to the mean of the data. Therefore we can replace Var(A s) with Es in equation 47-17 and Var(A r) with Et ... 

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

At any one instant, only a very small proportion of the total number of unstable nuclei in a radioactive source undergo decay. A Poisson distribution which expresses the result of a large number of experiments in which only a small number are successful, can thus be used to describe the results obtained from measurements on a source of constant activity. In practical terms this means that random fluctuations will always occur, and that the estimated standard deviation, 5, of a measurement can be related to the total measurement by ... 

Figure 4.10. Differences between Eqs. (4.10) from the single Gaussian distribution of Eq. (4,S) with R = 0.25, 1 is D -Dg. 2 is Dj-Dg, and 3 is D3-DG. D4--DG is so close to zero thai it does not even show up on this plot. The curves labeled "Poisson Noise" represent one standard deviation SPC decay data. All functions, if over] aid, are essentially indistinguishable. (Adapted from Ref. 55.)...

In LSC measurements precautions are required to avoid impurities which may cause scintillation quenching. Since radioactive decay is random and is described with the Poisson distribution, the standard deviation for a given count, C, is equal to C1/2. 

The Poisson distribution gives the probability of getting the result v in an experiment in which we count events that occur at random but a definite average rate //. The standard deviation of the Poisson... 

The standard deviation of the Poisson distribution is equal to xl/2. This has profound consequences in designing experiments involving radioactivity and light intensity. 

Like most natural events, radioactive decay is not a uniform function. Consequently, the term half-life is meant to describe the value that would result if an infinite number of half-life measurements were made and the average calculated. Individual decays, however, follow a Poisson distribution, i.e.. the standard deviation is equal lu the square root of the number of observed decay events. This fact enables the experimenter to calculate the probable accuracy of his result, assuming no instrumentation inaccuracy. 

Transformation based on square root from data X = /X is applied when the test values and variances are proportional as in Poisson s distribution. If the data come from counting up and the number of units is below 10 transformation form X --fX + 1 and text X =s/X + /X I 1 is used. If the test averages and their standard deviations are approximately proportional, we use the logarithm transformation X =log X. If there are data with low values or they have a zero value, we use X =log (X+l). When the squares of arithmetical averages and standard deviations are proportional we use the reciprocal transformation X =l/X or X =1/(X+1) if the values are small or are equal to zero. The transformation arc sin [X is used when values are given as proportions and when the distribution is Binomial. If the test value of the experiment is zero then instead of it we take the value l/(4n), and when it is 1, l-l/(4n) is taken as the value and n is the number of values. Transforming values where the proportion varies between 0.30 and 0.70 is practically senseless. This transformation is done by means of special tables suited for the purpose. 

Clearly these large fluctuations are due to cyclic variations not turbulent fluctuations. The dashed curve is an attempt to remove this cyclic variation effect by using the most probable density value as the mean value of a normal distribution. The standard deviation of the distribution is determined from fitting the data to the side of the new mean that has not been distorted by flame arrival. The reduction of the apparent fluctuations near the flame arrival crank angle is dramatic. Both curves of Figure 5 have had the Poisson statistical fluctuations subtracted. 

In absorption spectrometry, <7i is usually fairly constant, and x1 fitting has no advantages. Typical examples of data with nonconstant and known standard deviations are encountered in emission spectroscopy, particularly if photon counting techniques are employed, which are used for the analysis of very fast luminescence decays [27], In such cases, measurement errors follow a Poisson distribution instead... 

It may be shown that the mean of the Poisson distribution is X and the standard deviation VX. Hence, in any set of observations conforming to the distribution, mean = (S.D.)2,... 

The first term gives the probability of 0 particles, the second the probability of 1 particle, the third the probability of 2 particles, etc. The standard deviation of this distribution is given by total number of squares under observation. 

Figure 8-2. Poisson distribution with a mean of 40 and a standard deviation of V40. The vertical axis is the probability of obtaining the x-axis value.

This is a continuous function for the experimental variables, which is used as a convenient mathematical idealisation to describe the distribution of finite numbers of results. The factor 1 /(ay/lji) is a constant such that the total area under the probability distribution curve is unity. The mean value is given by p and the variance by a2. The variance in the Gaussian distribution corresponds to the standard deviation s in Eqn. 8.3. Figure 8-3 illustrates the Gaussian distribution calculated with the same parameters used to obtain the Poisson distribution in Figure 8-2, i.e. a mean of 40 and a standard deviation of V40. It can be seen that the two distributions are similar, and that the Poisson distribution is very dosely approximated by the continuous Gaussian curve. 

There are two sources of error that have to be taken into account when assessing the experimental error in a measured intensity. There are errors that arise from the random fluctuations in the detection system these errors follow a Poisson distribution and are proportional to the square root of the measured value (the count, hence the term counting statistics ). The total error in an intensity, represented by the estimated standard deviation (e.s.d.) cr(I), is approximated by counting statistics, and the second source of error, the instrumental uncer-... 

The o of a Poisson distribution can be shown to be equal to (N)1 2 where N Is the number of particles counted. An experimental standard deviation (S) for counts In all channels less than or equal to a, was chosen as the criterion for number of samples to Indicate that the error was due to counting statistics rather than any other experimental parameter. The results are presented In Table I and a channel size Identification In Table II. It can be seen that the above criterion was met when the number of samples was at least 6. It should be understood that when the total number of particles Is small. Increased counting times and number of samples are needed to attain statistical significance. 

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