Cumulative distribution function, defined

Anotlier fimction used to describe tlie probability distribution of a random variable X is tlie cumulative distribution function (cdf). If f(x) specifies tlie pdf of a random variable X, tlien F(x) is used to specify the cdf For both discrete and continuous random variables, tlie cdf of X is defined by ... 

Finally, the cumulative distribution function G(X) is defined as the integral function of the differential distribution function g(X) ... 

FIGURE 1.8 Probability density function (PDF) (left) and cumulative distribution function (right) of the normal distribution cr2) with mean /a and standard deviation cr. The quantile q defines a probability p. 

Marsh (1988), Cashman and Marsh (1988), and Cashman and Ferry (1988) investigated the application of crystal size distribution (CSD) theory (Randolph and Larson, 1971) to extract crystal growth rate and nucleation density. The following summary is based on the work of Marsh (1988). In the CSD method, the crystal population density, n(L), is defined as the number of crystals of a given size L per unit volume of rock. The cumulative distribution function N(L) is defined as... 

Next, we define a parallel set of NPD function in continuous flow recirculating systems. We restrict our discussion to steady flow systems. Here, as in the case of RTD, we distinguish between external and internal NPD functions. We define fk and 4 as the fraction of exiting volumetric flow rate and the fraction of material volume, respectively, that have experienced exactly k passages in the specified region of the system. The respective cumulative distribution functions, and /, the means of the distributions, the variances, and the moments of distributions, parallel the definitions given for the batch system. 

In an unmodified Monte Carlo method, simple random sampling is used to select each member of the 777-tuple set. Each of the input parameters for a model is represented by a probability density function that defines both the range of values that the input parameters can have and the probability that the parameters are within any subinterval of that range. In order to carry out a Monte Carlo sampling analysis, each input is represented by a cumulative distribution function (CDF) in which there is a one-to-one correspondence between a probability and values. A random number generator is used to select probability in the range of 0-1. This probability is then used to select a corresponding parameter value. 

The dissolution process can be interpreted stochastically since the profile of the accumulated fraction of amount dissolved from a solid dosage form gives the probability of the residence times of drug molecules in the dissolution medium. In fact, the accumulated fraction of the drug in solution, q (t) /goo, has a statistical sense since it represents the cumulative distribution function of the random variable dissolution time T, which is the time up to dissolution for an individual drug fraction from the dosage form. Hence, q (t) /q can be defined statistically as the probability that a molecule will leave the formulation prior to t, i.e., that the particular dissolution time T is smaller than t ... 

Because t appears in the integration limits of these two expressions, Equations (13-12) and (13-13) are both functions of time. Danckwerts defined Equation (13-12) as a cumulative distribution function and called it F(t). We... 

Before discussing our method for determining particle size, it is necessary to briefly review the definition of size distribution. If all particles of a given system were spherical in shape, the only size parameter would be the diameter. In most real cases of irregular particles, however, the size is usually expressed in terms of a sphere equivalent to the particle with regard to some property. Particles of a dispersed system are never of either perfectly identical size or shape A spread around the mean distribution) is found. Such a spread is often described in terms of standard deviation. However, a frequency function, or its integrated (cumulative) distribution function, more properly defines not only the spread but also the shape of such a spread around the mean value. This is commonly referred to as the particle size distribution (PSD) profile of the dispersed sample. 

The cumulative distribution function N Dp) for a lognormally distributed aerosol population is given by (8.39). Defining the normalized cumulative distribution... 

In financial mathematics random variables are used to describe the movement of asset prices, and assuming certain properties about the process followed by asset prices allows us to state what the expected outcome of events are. A random variable may be any value from a specified sample space. The specifica-ti(Mi of the probability distribution that appUes to the sample space will define the frequency of particular values taken by the random variable. The cumulative distribution function of a random variable X is defined using the distribution function yo such that Pr Xdiscrete random variable is one that can assume a finite or countable set of values, usually assumed to be the set of positive integers. We define a discrete random variable X with its own proba-bdity function p i) such that p i)=Pr X = /. In this case the probabiUty distribution is... 

The probability density function, fit), is defined as the probability of failure in any time interval df. The cumulative distribution function, F(t), is the integral of/(f). 

The normalising factor is L S(y) 0.02 + 0.15 + 0,315 +. .. + 0i)15 = 4.35 and the cumulative distribution function is obtained by dividing each of these figures by 4.35 and adding cumulatively. The resulting distribution is shown in Fig. 6.6. A measure of the safety of the column is the probability < 1 ] = 0.715. This means the probability that working stress is less than, or equal to the permissible stress is less than or equal to 0.715. The probability of failure in this defined limit state is 1 — 0.715 = 0.282. Of course, in a real example this figure would be much smaller. 

F(x) is thus defined for all real numbers. F(x) is also called the cumulative distribution function. If F(x) is differentiable, which normally is the case, we obtain its probability density function (pdf)... 

Modeling. In order to carry out the analysis of the nature of the operational phenomena in facilities and equipment, it is very useful to use statistics as a support for the quantification of the parameters. The phenomena s historical behavior is characterized based on operation and failure periods that have occurred since the commissioning time. The conditions that characterize the equipment operational time data are so numerous that it is not possible to say when exactly the next failure will occur. However, it is possible to express which will be the probability that the equipment is in operation or out of service at any given time. These times are associated with a cumulative distribution function of the random variable, which is defined as the addition of the probabilities of possible values of the variable that are lower or equal to a preset value. The mentioned random variable is constituted by the operating times and downtime of equipment or system in a given period. For its parameterization Weibull distribution is very appropriate as it is very effective and relatively simple to use in the reliability evaluation of a system by quantifying the probability of failure in the performance of the system s duties from the failure probabilities of its components based on the operation times. There are three different parameters ... 

The function E(t) is defined as function of RTD which represents the age distribution function of molecules in the fluid element derived from the cumulative distribution function. 

That both open and closed cracks contribute to modulus reduction suggests that an active crack density should be defined that describes the density of cracks participating in modulus reduction. The most transparent approach to define this is to use the cumulative distribution function Mol the distribution described in Figure 4 to determine the fraction of the closed cracks that are contributing to modulus reduction. 

The system fault probability function Ft or cumulative distribution function (CDF) of time to system failure is defined on [0, oo) - [0, 1], The corresponding probability density function (pdf) is defined on [0, oo) - [0, oo). The mean value of (1 - Ft) is the mean time to system failure (MTTSFt). 

Consider a life-testing experiment in which n identical units are put on test. Let Y, Y2, -, X denote the respective lifetimes from a population with cumulative distribution function F x) and probability density function f x). Let A i < X2-,n < < denote its order statistics. We define combined HCS as follows. 

Define the X axis as the probability that the buyer s value exceeds a certain value, 1 — Gi vi) = q, and the Y axis as value v. For each buyer i, graph the inverse of her cumulative distribution function Gi (where Gi ai) = 0, Gi bi) = 1) (see Figure 3.3). This represents the buyer s demand curve. The buyer s revenue is qvi, where Vi = G 1 — q). From the demand curve for each buyer, we may compute the buyer s marginal revenue as follows, i.e.,... 

Estimations of probability defined with help of cumulative distribution function can be applied for investigation within some area where maximum a and minimum b values of analyzed factor are established. In this case as hazardous parameter assess an interval probability Pab of karst forms development. Accuracy of estimation with using interval probability Pab depends on the sizes of study area and homogeneity of natural conditions within its borders. Let s notice, such approach to assessment of probability is widely applies in reliability theory of systems. 

Notice that this function was defined earlier for particle volume (Equation (2.3.5)). The function is clearly a cumulative distribution function because it is monotone increasing and approaches unity at infinite particle size, as it should. For continuous number density ffx, t) we may write... 

Discrete and continuous variables and probability distributions From Clause 5.3.3 of Chapter I, we get the probability mass function and cumulative distribution functions. For a single dimension, discrete random variable X, the discrete probability function is defined by/(xi), such that/(xi) > 0, for all xie R (range space), and f xi) = F(x) where F(x) is known as cumulative... 

For a continuous probability function, P x), the cumulative distribution function, Fx x), (cdf) is defined as... 

We now define a cumulative distribution function F(t) as the fraction of the molecules leaving the system with residence time t or less, or mathematically ... 

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