Probability density curve, table

The integral, which can be obtained from tables of error functions, has the value of 0 to 1/2 for x = 0 to oo and similarly for x 0 to — oo. This merely shows that whatever one has lost from one side of the boundary will have been gained on the other side. The plot of concentration gradient against x (obtained from refractive index changes) gives an integral probability density curve... 

The values of the error function erf(kA/f2) are given in Table 4.2. The reverse relation of A multiples relative to same most used values of the area under the probability density curve are given in Table 4.3. 

The probability-density function for the normal distribution curve calculated from Eq. (9-95) by using the values of a, b, and c obtained in Example 10 is also compared with precise values in Table 9-10. In such symmetrical cases the best fit is to be expected when the median or 50 percentile Xm is used in conjunction with the lower quartile or 25 percentile Xi or with the upper quartile or 75 percentile Xij. These statistics are frequently quoted, and determination of values of a, b, and c by using Xm with Xi and with xp is an indication of the symmetry of the curve. When the agreement is reasonable, the mean values ofo so determined should be used to calculate the corresponding value of a. 

Probability is expressed by a number between 0 and 1 that represents the chance that an event will occur. A probability of 1 means the event will definitely occur. A probability of 0 means the event will never occur. The probability or chance of occurrence is also expressed as a percentage between 0 and 100%. The probability function, P x), is also referred to as the probability density function, PDF x), or the cumulative probability function therefore, P x) = PDF x). The term probability function will be used throughout this book. The curve for a probability function P(x) of a normal distribution with a mean of p and a standard deviation of o can be mathematically described by Equation 6.1, as discussed before. The curve for a cumulative distribution function literally reflects the cumulative effect. The cumulative distribution function, D x), of a normal distribution is defined by Equation 6.5. It calculates the cumulative probability that a variate assumes a value in the range from 0 to X. Figure 6.3 is a plot of the cumulative distribution function curve from the data in Table 6.1. 

The value of q is multiplied by the tabulated values of the highest density regions for the degrees of freedom, and an appropriate confidence level (Table 26-4). The distribution curve (the posterior density of a) is skewed so that the interval is larger on one side of the standard deviation than on the other, with low values of the standard deviation less probable than high values. 

Figure 1 H + H2, J = 0. (a) Cumulative reaction probability. The solid curve is a spline fit to the accurate quantal results, and the dashed curve was obtained by integrating the synthetic density in b. (b) Density of reactive states. The solid curve is obtained by analytically differentiating a cubic spline fit to the accurate quantum mechanical CRPs. The heavy dashed curve is the fit of Eqs. (14) and (15). The arrows are positioned at the fitted values of T, and the feature numbers and assignments above the arrows correspond to Table 2. (Reprinted with permission from Ref. 8, copyright 1991, American Chemical Society.)...

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