Joint Event Probabilities

To obtain the nn-pair correlation we need the probability of the joint event (s, = 1, = 1) i.e., both site i and i + 1 being occupied. This may be obtained by following steps similar to those for P s- = 1). Since our system is cyclic, the nn-pair distribution is independent of index i, hence we write... 

Therefore, the probability of the desired joint event may be written as... 

The link between the probabilistic transfer model and retention-time distribution model may be explicitly demonstrated by deriving the conditional probability implied in the one-compartment probabilistic transfer model. We look for the probability, S (a + A a), that a particle survives to age (a + A a). Clearly, the necessary events are that the particle survives to age a, associated with the state probability S (a) AND that it remains in the compartment during the interval from a to (a + A a), associated with the conditional probability [1 — hAa, where h is the probabilistic hazard rate. Therefore, the probability of the desired joint event may be written as... 

Let us introduce the concept of conditional probability. We denote the joint probability of events A and B by p(A,B) and the conditional probability of occurrence of event A given event B by p(A B). Thus, we have... 

Joint probability. The probability of events occurring together. 

By using the definition of conditional probability in Equation (2.1), the probability of the joint events can be obtained ... 

For instance, if = 4, Xn = X12 +X123 -FX124 -FX1234. Equation (3) originates with the inclusion-exclusion principle for the probability of joint events ), because Ai2...m affects all systems states for which at least one component /( < / < ) is still operating before the multiple failure ... 

Figure 12.6 The concept of joint probability of events [source Spurr and Bonini (1973)].

What are the probability formulas for a joint event, compound event, and simple event probability ... 

Joint Event Probability the probability of an event in which the researcher asks the question What is the probability of event A and event B occurring at the same time ... 

This survey of IT probes of chemical bonds continues with some rudiments on the entropic characteristics of the dependent probability distributions and information descriptors of a transmission of signals in communication systems [3,4,7,8]. For two mutually dependent (discrete) probability vectors of two separate sets of events a and b, P(a) = P(fl) =Pi) =p and P(b) = P(pp = qj = q, one decomposes the joint probabilities of the simultaneous events a/ b = [a,Air) into these two schemes, P afdb) = P(fli/ b) = Jt,y] = 31, as products of the margin probabilities of events in one set, say a, and the corresponding conditional probabilities P( la) = [P(/li) = 3t,y/Pj] of outcomes in set b, given that events a have already occurred [jty=p (j i). The relevant normalization conditions for the joint and conditional probabilities then read ... 

The mutual information of an event with itself defines its self-information I(i i) = I(t) = log[P(i i)/pi = -logPi, since P(i i) = 1. It vanishes when p = 1 (i.e., when there is no uncertainty about the occurrence of aj) so that the occurrence of this event removes no uncertainty, hence conveying no information. This quantity provides a measure of the uncertainty about the occurrence of the event (i.e., the information received when the event occurs). Shannon entropy can thus be interpreted as the mean value of self-information in all individual events S(p) = pj i). One similarly defines the average mutual information in two probability distributions as the n-weighted mean value of the mutual information quantities for the individual joint events ... 

It should be also observed that the average mutual information is an example of KL entropy deficiency measuring the missing information between the joint probabilities P(aAft) = 71 of the dependent events a and b, and the joint probabilities P (aAfc) = = p q of the independent joint events I(p q) = A5( iIji°). The average... 

This type of chance constraints is named as joint chance constraints. Where Pr - represents the probability of event established in , a point x is feasible if and only if the probability of event gj x, ) < 0,j = 1,2,...,/ is no less than a, which means the probability of constraint violation is less than (1 — a). 

Joint Probability. The joint probability of events. 4 and B is the probability that both events. 4 and B occur. The Joint probability is expressed by the notation p(A and B), or more concisely by p(AB). 

General Multiplication Rule (Bayes Rule). If outcomes A and B occur with probabilities piA) and p(B), the Joint probability of events 4 and B is... 

The probability that event i is first is p(i). Then the conditional probability that event j is second is p(j)lii - pii)). The joint probability that i is first, j is second, and k is third is... 

Probability provides a classical model for dealing with uncertainty (Halpem 2003). The basic elements of probability theory are a) random variables and b) events, which are appropriate subsets of the sample space Q. A probabilistic model is an encoding of probabilistic information that allows the probability of events to be computed, according to the axioms of probability. In the continuous case, the usual method for specifying a probabilistic model assumes a full joint PDF over the considered random variables. 

This ensemble over J7 ,Fn can also be considered as a triple product ensemble, UnVnZn, where Zn is an ensemble consisting of the events zt corresponding to error, and ze corresponding to no error. The joint probabilities in this triple product ensemble are... 

The joint velocity, composition PDF is defined in terms of the probability of observing the event where the velocity and composition random fields at point x and time t fall in the differential neighborhood of the fixed values V and ip ... 

Another question may be asked suppose / have observed an event at ta, what is the probability distribution w(01 ta) of the time I have to wait until the next event The joint probability for having one event between ta — dta and ta, and one in (tb, tb + dtb) with no events between them is... 

Subdivide the total volume Q into cells A and call nk the number of particles in cell X. The cells must be so small that inside each of them the above mentioned condition of homogeneity prevails. Let P( nk, t) be the joint probability distribution of all nk. At t + dt it will have changed because of two kinds of possible processes. Firstly, the nk inside each separate cell X may change by an event that creates or annihilates a particle. In the master equation for P( nk, t) this gives a corresponding term for each separate cell. 

Notice that the arguments for the joint probabilities are ordered such that t < t2 < tn, so the order of events should be read from the right to the left. To continue we must somehow truncate the series of higher-order joint probability densities. The simplest case (often referred to as a purely random process) is one in which the knowledge of P(y,t) suffices for the solution of the problem. In particular,... 

To obtain equations for the state probabilities, write the equation for the state probability at t + At as the sum of joint probabilities for all the mutually exclusive events that enumerate all the possible ways in which a particle starting in i at 0 could pass through the various compartments at time t AND end up in j at t + At. These joint probabilities can be expressed as the product of a marginal by a conditional probability. The state probability p (t) that a given particle starting in i at time 0 is resident in compartment s at time t plays the role of the marginal and the transfer probability hSj (t) At that a given particle resident in compartment s at time t will next transfer to compartment j, i.e., at time t + At plays the role of the conditional probability... 

Prompted by these considerations, Gillespie [388] introduced the reaction probability density function p (x, l), which is a joint probability distribution on the space of the continuous variable x (0 < x < oc) and the discrete variable l (1 = 1,..., to0). This function is used as p (x, l) Ax to define the probability that given the state n(t) at time t, the next event will occur in the infinitesimal time interval (t + x,t + x + Ax), AND will be an Ri event. Our first step toward finding a legitimate method for assigning numerical values to x and l is to derive, from the elementary conditional probability hi At, an analytical expression for p (x, l). To this end, we now calculate the probability p (x, l) Ax as the product po (x), the probability at time t that no event will occur in the time interval (t, t + x) TIMES a/ Ax, the subsequent probability that an R.i... 

This condition recognizes a general category of these input-dependent probabilities [p(k i) and p(k i0) as conditional probabilities of two-orbital events, that is, the joint probabilities per unit probability of the specified input p(k i) = p(k i) and p(k i°) = p(k zfl). However, it should be emphasized that these probabilities are also conditional on the molecule as a whole, since... 

Finally, we adopt a notation involving conditional averages to express several of the important results. This notation is standard in other fields (Resnick, 2001), not without precedent in statistical mechanics (Febowitz et al, 1967), and particularly useful here. The joint probability P A, B) of events A and B may be expressed as P A, B) = P A B)P B) where P B) is the marginal distribution, and P A B) is the distribution of A conditional on B, provided that P B) 0. The expectation of A conditional on B is A B, the expectation of A evaluated with the distribution P(A B) for specified B. In many texts (Resnick, 2001), that object is denoted as E(A B) but the bracket notation for average is firmly established in the present subject so we follow that precedent despite the widespread recognition of a notation (A B) for a different object in quantum mechanics texts. 

Cao has described the event echo [49] as difference between a joint (correlated) probability to detect photons in time p(ti,T2) and a disjoint (uncorrelated) propability p(ti)p(t2). The difference distribution function is given by... 

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