Probability cumulative distribution function

The probability density of the normal distribution f x) is not very useful in error analysis. It is better to use the integral of the probability density, which is the cumulative distribution function... 

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 ... 

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

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 ... 

After a point estimation is performed, the question is how much the deviation of the estimate is likely to be from the still unknown parameter. As it was pointed out by Mikhail (1976), it is only possible to estimate the probability that the true value of the parameter is likely to be within a certain interval around the estimate if the cumulative distribution function F(x) of the random variable is given. 

Figure 4.1 Relationship between the probability density function f x) of the continuous random variable X and the cumulative distribution function F(x). The shaded area under the curve f(x) up to x0 is equal to the value of f x) at x0. 

Because F(x) is non-decreasing, its derivative /(x) is non-negative. Conversely, if fi is the domain ] — oo, + oo[, the cumulative distribution function F(x) relates to the probability density function /(x) through... 

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. 

The normal distribution, A Y/l, o 2), has a mean (expectation) fi and a standard deviation cr (variance tr2). Figure 1.8 (left) shows the probability density function of the normal distribution N(pb, tr2), and Figure 1.8 (right) the cumulative distribution function with the typical S-shape. A special case is the standard normal distribution, N(0, 1), with p =0 and standard deviation tr = 1. The normal distribution plays an important role in statistical testing. 

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. 

The subsequent calculations would probably be a lot harder in this case. There are also various other approaches based on density ratios (bounded density distributions), 8-contamination models, mixtures, quantile classes, and bounds on cumulative distribution functions. See Berger (1985,1994) for an introduction to these ideas. 

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. 

Probability bounds analysis takes as inputs structures called p-boxes, which express sure bounds on a cumulative distribution function. One p-box is depicted in Figure 6.4. 

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. 

Parameter Two distinct definitions for parameter are used. In the first usage (preferred), parameter refers to the constants characterizing the probability density function or cumulative distribution function of a random variable. For example, if the random variable W is known to be normally distributed with mean p and standard deviation o, the constants p and o are called parameters. In the second usage, parameter can be a constant or an independent variable in a mathematical equation or model. For example, in the equation Z = X + 2Y, the independent variables (X, Y) and the constant (2) are all parameters. 

Probability box A kind of uncertain number representing both incertitude and variability. A probability box can be specified by a pair of functions serving as bounds about an imprecisely known cumulative distribution function. The probability box is identified with the class of distribution functions that would be consistent with (i.e., bounded by) these distributions. 

The random variable x has a continuous distribution fix) and cumulative distribution function F(x). What is the probability distribution of the sample maximum (Hint In a random sample of n observations, x, x2,. .., x , if z is the maximum, then every observation in the sample is less than or equal to z. Use the cdf.)... 

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. 

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 ... 

For some variables, for example, the relative collision velocity, the cumulative distribution function does not have closed form, and then a third Monte Carlo method must be adopted. Here, another random number R is used to provide a value of v, but a decision on whether to accept this value is made on the outcome of a game of chance against a second random number. The probability that a value is accepted is proportional to the probability density in the statistical distribution at that value. The procedure is repeated until the game of chance is won, and the successful value of v is then incorporated into the set of starting parameters. 

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). 

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

Plotting the ECDF versus TCDF, for a given probability, generates the linear version of a cumulative distribution function, a quantile-quantile (QQ) plot. Each cumulative probability value yields a pair of order statistics (one from each CDF) that form a point in the QQ plot. Quantile-quantile plots are valuable tools for distinguishing differences in shape, size, and location between spectral clusters. Two similar clusters of spectra will demonstrate a linear QQ plot (Fig. 1). Breaks and/ or curves in the QQ plot indicate that differences exist between the groups (Fig. 2). 

The mean and standard deviation of the normal distribution are T and a, respectively. Since the normal distribution is designed for continuous data, the cumulative distribution function is more practical than the probability density function. For a particular data population, the cumulative distribution [2] is as follows ... 

Making this substitution into Equation (3.6) or (3.7) reduces the generic normal distribution to one with mean 0 and standard deviation 1, collapsing all possible normal distributions onto a standard curve. Tabulated values of the cumulative distribution function F are usually presented in terms of the transformation variable z. Sample values of F(z) are presented in Table 3.2. Microsoft Excel contains an intrinsic function, NORMSDIST, that produces the cumulative probability for a standard normal variable z given as its argument. A companion function, NORMSINV, outputs the z value for a given F(z). The Microsoft Excel manual or the electronic help files [5, 6] provide command syntax and usage examples. 

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... 

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