Discrete Probabilities

The percentages chosen are often denoted as the p85, p50, pi 5 values. Because they each approximately represent one third of the distribution, their discrete probabilities may each be assigned as one third. This approximation is true for a normal (or symmetrical) PDF. 

Many companies choose to represent a continuous distribution with discrete values using the p90, p50, plO values. The discrete probabilities which are then attached to these values are then 25%, 50%, 25%, for a normal distribution. 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

Property 1 indicates tliat tlie pdf of a discrete random variable generates probability by substitution. Properties 2 and 3 restrict the values of f(x) to nonnegative real niunbers whose sum is 1. An example of a discrete probability distribution function (approaching a normal distribution - to be discussed in tlie next chapter) is provided in Figure 19.8.1. 

An order density is a demand density 5 with 5(0) = 0. The number of orders per interval can be described by a discrete density function t] with discrete probabilities defined for nonnegative integers 0,1, 2, 3, —The resulting (t], 8)-compounddensity Junction is constructed as follows A random number of random orders constitute the random demand. The random number of orders is r -distributed. The random orders are independent from the number of orders, and are independent and identically 5-distributed. 

The conditional service density derived from a continuous demand density is continuous almost everywhere but has one exceptional point which carries a discrete probability mass The probability that the whole inventory s goes to service is the integral of the conditional demand density from s to oo. In other words, the service is s if the demand is s or above. 

This model is directly derived from the Langmuir isotherm. It assumes that the adsorbent surface consists of two different types of independent adsorption sites. Under this assumption, the adsorption energy distribution can be modeled by a bimodal discrete probability density function, where two spikes (delta-Dirac functions) are located at the average adsorption energy of the two kinds of sites, respectively. The equation of the Bilangmuir isotherm is... 

Gordon Discrete Probability. Hairer/Wanner Analysis hy Its History. Readings in Mathematics. 

This case has been dealt with for continuous (rather than discrete) probability distributions in an earlier paper.27 Also, it is worth noting that the physics of this process are similar to the process of coagulation in atmospheric physics.37... 

For a discrete probability distribution, we can define a function f which gives the probability P of getting the outcome A=x, where A is the discontinuous variable. 

Neurotransmitter release is not assured in response to synaptic stimulation. Rather, the process of vesicle fusion for individual release-competent vesicles is probabilistic. This process confers a discrete probability (between 0 and 1) that a given synapse will release neurotransmitter after an action potential (the synaptic release probability). For the majority of synapses in the central nervous system, the release probability at a deflned synaptic contact is below 0.3, which leads to the often-quoted statement that the release process is reliably unreliable (1). Despite this fact, it has been demonstrated that some central nervous system synapses (in a variety of brain regions) do exhibit release probabilities as high as 0.9 (2-4). This higher synaptic release... 

A.4.11 Discrete probability distributions model systems with finite, or countably infinite, values, while a continuous probability distribution model systems with infinite possible values within a range. 

A simple example of a discrete probability distribution is the process by which a single participant is assigned the active treatment when the event "active treatment" is equally likely as the event "placebo treatment." This random process is like a coin toss with a perfectly fair coin. If the random variable, X, takes the value of 1 if active treatment is randomly assigned and 0 if the placebo treatment is randomly assigned, the probability distribution function can be described as follows ... 

Since distributions describing a discrete random variable may be less familiar than those routinely used for describing a continuous random variable, a presentation of basic theory is warranted. Count data, expressed as the number of occurrences during a specified time interval, often can be characterized by a discrete probability distribution known as the Poisson distribution, named after Simeon-Denis Poisson who first published it in 1838. For a Poisson-distributed random variable, Y, with mean X, the probability of exactly y events, for y = 0,1, 2,..., is given by Eq. (27.1). Representative Poisson distributions are presented for A = 1, 3, and 9 in Figure 27.3. 

This example demonstrates the simple method by which a discrete probability distribution can be determined. 

Having related the discrete and continuous probability we can likewise relate the probability current jk t) of the discrete system to the probability current J x,t) of the continuous envelope description. According to the relation between the discrete and continuous probability eq. (2.20), the discrete probability current t) from fc to fc + 1 is equal to the continuous probability current t x = k + (see Fig. 2.7),... 

In this section we describe the six discrete probability distributions and five continuous probability distributions that occur most frequently in bioinformatics and computational biology. These are called univariate models. In the last three sections, we discuss probability models that involve more than one random variable called multivariate models. 

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