Probability Mass Function

A compound is flagged as either active or inactive on the basis of the activity threshold determined as in Section 6.2.2. The probability that the observed number of actives x occurs in a cluster of size n given a background hit rate p (expressed as a fraction) is determined by the binomial probability mass function [38], shown as follows ... 

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... 

Binomial distribution. The simplest common probability model used for binary outcomes from a series of independent trials. (Where tried is to be understood in the sense of the first meaning of the definition.) If n is the number of trials, 0 is the probability of success in an individual trial and X is the number of successes, then the probability mass function of the binomial distribution is... 

Probability mass function. A function which assigns a probability to values of a random variable. Used for discrete variables as opposed to probability density function, which is used for continuous ones. An example is given in the entry under Poisson distribution. 

Note that it is often convenient to maximize the log-likelihood function, a monotone function of the likelihood whose maximum will correspond to the maximum of the original likelihood function. If our variables are discrete, we instead use the multinomial distribution whose probability mass function is as follows ... 

To illustrate, consider the problem summarized in Table 3. There are three risky, independent, end-of-year cash flows the means and variances of their respective probability functions are given in columns (2) and (3) of Table 3. Project life, N, is also a random variable, with probability mass function as shown in columns (6) and (7) of the table. A 10% discount rate is assumed. Determination of the expected present worth, based on Eq. 17, is summarized in the table. Here, fXp = — 82.64. The variance of present worth, 0-2, based on Eqs. 18 and 19, may be shown to be... 

Generally, to calculate the probability of a certain number of successes in a series of trials, where the probability of success is constant, and where the trials are independent, it is appropriate to use the binomial probability mass function ... 

The potential for reduced size and complexity is important if BN models are to be applied on a grand scale, modelling the complete set of loss events faced by a financial institution. Because each node in the network needs to be assigned a (conditional) density or probability mass function a simple structure may provide more efficient modelling as the required input is reduced. Hence the model suggested in this paper may potentially reduce the effort needed in constructing the BN model. 

NBMs have most widely been used with sparse extended connectivity fingerprints [5, 7,12, 14, 22], Other descriptors also have been explored, such as 2D pharmacophore triplets [23] and atom types [24], Molecular fingerprints were also combined with continuous descriptors [25]. Although widely apphed naive Bayesian implementations bin continuous descriptors, the Bayesian framework naturally allows for incorporation of continuous descriptors characterized by a probabihty density, rather than a probability mass function, as in the binned case. 

For example, occurrence can be modeled as a Poisson distribution under the assumption that the disruptive events are not correlated. Equation 7.6 gives the probability mass function of the Poisson distribution ... 

For discrete variable X, there exists a function/(x) such that f(x) = P(X = x), x being the general value of X. Function of this kind satisfying the following conditions is called probability mass function of X. 

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... 

The situation, however, can be rectified with additional assumptions, one of which was used to determine the value for/and therefore The value for / however, could vary from 1.84 to 1.92 depending upon the specifics of the potential between the adsorbent surface atoms and the adsorbate molecules. The value of 1.84 has been used as most reasonable, but this could introduce an error of as much as 5%. It does, however, leave the theory free from the burden of needing to know the specifics of the surface composition. The assumption of the LJ 6-12 assumption for both adsorbate and adsorbent is retained. A second assumption is that within the LJ 6-12 potential only the ground quantum state of vibration is important. This is an extremely justifiable assumption since most adsorption measurements are performed at room temperature or below. Some simple calculations indicate for most cases that the second state is occupied by much less than a part per million. (Spectroscopists consistently use this assumption almost without thinking about it.) The ground state for vibration is represented by the first Hermite polynomial (//q), which is conveniently identical to the probability mass function (PMF) or Gaussian ... 

For the curve fitting, it will be assumed that there is a distribution of energies, E s, and a distribution of pore sizes. Furthermore, some of the surface area is not inside the pores and is referred to as external. The pore radius is reflected in a cutoff in the standard curve or in terms of x there is a mean value (x for which the standard curve in the pores is terminated. The probability mass function (PMF) distribution will be used with the standard deviations for energy and pore size. Any reasonable distribution could be used and the parameters expanded, for example to include skewness, etc., but usually the experimental data would not justify this. Thus there are six parameters ... 

To find the desired probability that an event occurs use a probability density function when we have continuous random variables or a probability mass function in the case of discrete random variables. In this text, f(x) is used to represent the probability mass or density function for either discrete or continuous random variables, respectively. We now discuss how to find probabilities using these functions, first for the continuous case and then for discrete random variables. To find the probability that a continuous random variable falls in a particular interval of real numbers, we have to calculate the appropriate area under the curve of f(x). Thus, we have to evaluate the integral of f(x) over the interval of random variables corresponding to the event of interest. This is represented by ... 

Alternatively, we may also sum up the individual values of the binomial probability mass function (binopdf) ... 

Consider the substances i in a product that has a demand of u. The supplier makes a sustainability investment in i. Assume the unit cost of i before and after the sustainability investment to be Ci and C2, respectively. Assume the demand enhancement of the product due to the improved substance i to be a random variable (mean and variance with probability mass function/( ). Clearly, the amount of i procured before the sustainability investment is u. Assume the purchased amount of improved substance i to be m -i- q. 

As mentioned earlier, the demand for healthcare depends on such factors as price, income, age, and preferences. In most cases, the need for healthcare arises only when the person is sick. The two major determinants of a sick person s use of healthcare services are the price of the service and household income. We define f p I, 6) as the probability mass function that a random individual will pay the amount p for the use of healthcare, where 7 and 6 are the income and health status of the individual, respectively. Note that 7 = 1 (7 [0,1]) is the highest income level, and G = G [0,1]) is the health level of the sickest person in the population. Therefore, the probability mass function of payment x by the population as a whole... 

Occurrence is the frequency of a certain disruptive event over a period of time (e.g., a year or several years depending on the event of interest). Tamhane and Dunlop (2000) state that the Poisson distribution is suitable to model the occurrence of rare events, unless they are correlated. The probability mass function of the Poisson distribution is shown in Equation 10.6. 

Cumulative distribution function of a discrete random variable Probability mass function of a discrete random variable Derivative of Gx(z) with respect to z... 

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