Cumulative probability function

Frequency analysis is an alternative to moment-ratio analysis in selecting a representative function. Probability paper (see Figure 1-59 for an example) is available for each distribution, and the function is presented as a cumulative probability function. If the data sample has the same distribution function as the function used to scale the paper, the data will plot as a straight line. 

E(f) may be integrated to any upper limit t to give the cumulative probability function F(t). This is illustrated in Fig. 2. 

A plausible estimate of the spatial extension of a hydrogenic orbital is the radius of a spherical boundary surface within which there is a high probability of finding the electron. In order to develop this criterion on a quantitative basis, it is useful to define a cumulative probability function FW(p) which gives the probability of finding the electron at a distance less than or equal to p from the nucleus ... 

Table 2.1.6. Cumulative probability functions for hydrogenic orbitals...

The cumulative probability functions listed in Table 2.1.6 can be used to give some measure of correlation between the various criteria for orbital size. Once a particular value for P is chosen, it can be equated to each of the expresssions in turn and the resulting transcendental equation solved numerically. The p values thus obtained for P = 0.50, 0.90, 0.95, and 0.99 are tabulated in Table 2.1.7. The significant quantites are the size ratios, with the Is p value as standard, which provide a rational scale of relative orbital size based on any adopted probability criterion. As the prescribed P value approaches unity, the size ratios gradually decrease in magnitude and orbitals in the same shell tend to converge to a similar size. 

The residence times in a continuous flow reactor have a distribution that can be characterized by any of a trio of functions. One of these is the cumulative probability function F(t), which is the fraction of exiting material that was in the reactor for a time less than t. 

A stochastic model of lignin structure and liquefaction was developed. Lignin was viewed as an ensemble of phenolic ringcomprising oligomers. Monte Carlo simulation of lignin structure, via the placement of random numbers on cumulative probability functions for the substitutes on each ring position, provided Np >... 

Here, S stands for the amount of damage or magnitude of loss, and F S) is its distribution function (cumulative probability function). 

The normal probability relationship and its familiar beU-shaped curve represent a totahty of data, all of the scores on a test, average soil resistivities, or all pit depths form the basis for the curve. Application of the cumulative probability function for an exponential extreme value distribution of a standard variate to practical situations requires statistically valid collection of data. A practical and consistent sample size must be selected and enough samples must be taken to attain reliable results. 

Figure 13,6 Cumulative probability function of the standard normal distribution...

Figure 21.9 Both graphs plot the cumulative probability function of all compounds (black line) and the cumulative probability function of the fragments (A squares, B circles) against the % effect seen in the HTS. In graph A, a...

The time to failure of specimens subjected to an accelerated lifetest is a random variable which is usually distributed according to a lognormal cumulative probability function (CPF). Once the shape and the values of the parameters characterising the CPF have been determined, it is possible to predict the failure rate of the sampled population as a function of the time in service." ... 

A number of statistical transformations have since then been proposed to quantify the distributions in pitting variables. Gumbel is given the credit for the original development of extreme value statistics (EVS) for the characterization of pit depth distribution [10]. The EVS procedure is to measure maximum pit depths on several replicate specimens that have pitted, then arrange the pit depth values in order of increasing rank. The Gumbel or extreme value cumulative probability function [f(x)] is shown in Eq (6.1), where A and a are the location and scale parameters, respectively. This probability function can be used to characterize data sets and estimate the extreme pit depth that possibly can affect a system. 

Figure 10.14 Cumulative Probability Function for Triangular Distribution...

Since the standard normal space is rotational symmetric, probability of failure can be directly obtained using the reliability index, Pj =<6(-y, where <6 is the standard normal cumulative probability function. 

Weibull distribution function is one of the widely used cumulative probability functions for predicting lifetime in reliability test [34]. This is because it can easily approximate the normal distribution, logarithmic normal distribution and exponential distribution functions. In addition, it is also possible to analyze data even when two or more failure modes are present at the same time. The cumulative probability F(t) of a failure system can be introduced just as Weibull distribution function based upon a weakest-link model [34], which is expressed as ... 

From the probability density function it is derived the cumulative probability function P x), that is the probability that an element of the population assumes a value less or equal to a given x. 

Fig. 4.13 Trend of the Weibull cumulative probability function for some values of the m exponent...

Using the Weibull cumulative probability function given by (4.37) Eq. 4.44... 

Assuming as cumulative probability function the Weibull expression P df) given by Eq. (4.45), then the expected value of will be... 

The three load histories constitute the unit block-program. Knowing that 1,000 work pieces will be built, calculate the number m of blocks that can be tolerated by the beam without fatigue failure with 99.9 % of probability of survival Ps — I — P, P being the cumulative probability function. The three moments at the fix end of the cantilever beam corresponding to the three loads are... 

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. 

---