Gaussian distribution root-mean-square value

There is a certain difficulty in the consideration of zero average tension. While the most probable end-to-end distance of an unstrained molecular chain is r = a JlNj3, and the root-mean-square value for r in a Gaussian distribution is = a %/5v"(Treloar93 p. 56), a tension in the chain exists for all values of r (loc. cit., p. 62), including , other than zero. We do not resolve the difficulty here, but for our purpose assume that the chain is under no tension different from that which would be found in a free chain of the same length. 

Find the mean value and the root-mean-square value of sin(x) for —oo < x < oo, assuming the standard normal (Gaussian) probability distribution... 

The root-mean-square error (RMS error) is a statistic closely related to MAD for gaussian distributions. It provides a measure of the abso differences between calculated values and experiment as well as distribution of the values with respect to the mean. 

We used p instead of = in Equation 5.37 because the exact numerical value depends on the definition of the uncertainties—you will see different values in different books. If we define At in Figure 5.13 as the full width at half maximum or the root-mean-squared deviation from the mean, the numerical value in Equation 5.37 changes. It also changes a little if the distribution of frequencies is not Gaussian. Equation 5.37 represents the best possible case more generally we write... 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

The Gaussian distribution has a symmetrical, bell shape and is sufficiently characterized by its mean (/jl) and variance (cr2). Its standard deviation (cr) is the square root of the variance. The symbols pi and cr refer to the mean and standard deviation of a population (i.e., the set of all possible measurements of a particular quantity). In practice, /x, and cr often are not known because a population is too large to sample in its entirety. A subset of measurements of a particular quantity represents a sample of a population. If the sample is drawn from a normal population, the parameters characterizing the sample distribution are the sample mean of the measurements or observations (y) and the sample standard deviation (s). Both of these parameters are calculated based on the values of the sample observations and the number of observations,... 

In other words, it is the square root of the mean of the squares of the deviations X — m. In distributions like the Gaussian distribution,t the standard deviation is not fixed by the value of the measurement it varies with the precision with which the experiments are made, being small in high-precision work and large in work of low precision. The Poisson equation, on the other hand, is not concerned with the precision of the experiments it relates to a situation where there is a natural spread of events. 

One important function is the Gaussian distribution, often called the normal distribution or bell curve which is defined by two parameters. The mean, //, describes its average value. The variance denoted describes whether the distribution is narrow or dispersed. The square root of the variance is called the standard deviation and is denoted o. The Gaussian distribution is shown in Figure 15.1 and defined by the equation... 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Thus the mean and variance of a gamma distribution are sufficient to determine its two parameters, a and /3. Note that the coefficient of variation (standard deviation divided by the mean) is equal to the square root of 1/or. The most probable value of k (the mode) occurs at (a - l)//3 if oc > 1, and as a T oo the gamma distribution itself becomes the gaussian or normal distribution for the variable, /8k.13... 

---