Cumulative Distribution Functions CDF

For the robust estimation of the pair potentials, some obstacles had to be overcome. There are a huge number of different triples (si, Sk,i — k), and to find densities, we needed a way to group them in a natural way together into suitable classes. A look at the cumulative distribution functions (cdf s) of the half squared distances Cjfc at residue distance d = i — k (w.l.o.g. >0), displayed in Figure 1, shows that the residue distances 8 and higher behave very similarly so in a first step we truncated all residue distances larger than 8 to 8. 

Figure 4.6 Shape of the Cumulative Distribution Function (CDF) for an arbitrary normal distribution with varying standard deviation (adapted from Carter, 1986)...

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 ... 

One way to introduce the data values into the estimation algorithm is to consider the conditional probability distribution of the unknown P( c), given the N data values used to estimate it Denote this conditional cumulative distribution function (cdf) by ... 

Time profiles in vitro and in vivo represent distribution functions in a mathematical and statistical sense. For example, a release profile Fj)(t) in vitro expresses the distribution of drug released at time t the corresponding probability distribution function (PDF) profile fo(t) characterizes the rate of release. Similarly, a plasma concentration profile fp(t) represents the distribution of drug in the plasma at any time t, i.e., absorbed but not yet eliminated its cumulative distribution function (CDF) equivalent FP(t) represents the drug absorbed and already eliminated. 

Species sensitivity distributions are sometimes fitted by minimizing the sum of squared deviations between the empirical cumulative distribution function (cdf) and the fitted cdf. 

FIG U RE 6.3 Two parametric classes of prior distributions having constant variance (left) or constant mean (right) shown as cumulative distribution functions (cdfs). The horizontal axis is some value for a random variable and the vertical axis is (cumulative) probability. 

Cumulative distribution function (CDF) The CDF is referred to as the distribution fnnction, cumulative frequency function, or the cnmnlative probability fnnction. The cumnlative distribution fnnction, F(x), expresses the probability that a random variable X assumes a value less than or eqnal to some valne x, F(x) = Prob (X > x). For continnons random variables, the cnmnlative distribution function is obtained from the probability density fnnction by integration, or by snmmation in the case of discrete random variables. 

Figure 3. The stair step curve is the detected sample cumulative distribution function (CDF) for the combined detectors. We have weighted the two detectors equally so that the height of an IMB detection is 12/8 the height of a Kamiokande detection (note we have included the count at. 686 seconds rejected by the Kamiokande group as being too close to their threshold). If millions of counts had been seen, the CDF would be smooth and directly proportional to the number luminosity emitted by the supernova.

Cumulative distribution function (cdf) for a random variable, say X, is a function, say F, such that for any value t, F t) is the probability that X is less than or equal to t. 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

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. 

Darling normality test [27], probability plot and cumulative distribution function (CDF) are eriteria that eould be used to eheek the normality of the data [25, 27 and 28],... 

Cumulative distribution function (cdf) For a random variable (X), this is a function (F) such that for any value t, F(t) is the probabihty that X is less than or equal to t. For example, if the random variable X is the margin of exposure, then the cumulative distribution function evaluated at 100, i.e. F(IOO), is the probability that the margin of exposure is less than or equal to 100, while F(IOOO) is the probability that the margin of exposure is less than or equal to 1000 (Sielken, Ch. 8). 

An example of inverse transformation is the exponential random variable X with mean 1/A and variance l i, which has cumulative distribution function (CDF)... 

Every possible outcome of a random variable is associated with a probability for that event occurring. Two functions map outcome to probability for continuous random variables the probability density function (pdf) and cumulative distribution function (cdf). In the discrete case, the pdf and cdf are referred to as the probability mass function and cumulative mass function, respectively. A function f(x) is a pdf for some continuous random variable X if and only if... 

The function Fix) is called the cumulative distribution function (cdf). The cdf has the following properties ... 

In Figure 1, we compare the complementary cumulative distribution functions (C(2DF) and cumulative distribution function (CDF) for P obtained directly from simulations using (2) or (4), with the approximate value of P using the linear expression P = kt. For the 100m scale, k is estimated using fracture density and flow porosity as k = 2Sj jn, whereas... 

Suppose that annual operating savings, A, is a discrete random variable with probabilities as given in Table 9. The associated cumulative distribution function (CDF), also given in the table, represents the probability that the annual operating savings will be less than or equal to some given vine. 

Event i Annued Savings A, Probability of Occurrence" P(Ad Cumulative Distribution Function (CDF) P(Ann. Savings < Aj)... 

CSGs (Constructive Solid Models), 182 C/S systems, see Client/server systems CTA, see Cognitive task analysis CTDs, see Cumulative trauma disorders CTP (capable-to-promise), 2046 CTS, see Ctupal tunnel syndrome Cuban Missile Crisis, 139 Culture. See also National culture Oigemizational culture and eilignment of technology/oiganizational structure, 956-961 safety, 959-961 Culture shift, 14, 16 Culture systems, 15-16, 1798 Cumulative distribution function (CDF), 2385-2386... 

To simplify the probability calculation of permuting coincidence patterns, it is assumed that the reliability properties of all systems can be characterised by the same cumulative distribution function (CDF). 

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