Definitions of conditions

Acne skin disease occurring in the presence of sebaceous glands 

Acute laryngotracheobronchitis viral croup Agranulocytosis a drastic reduction in the white blood cell count 

Allergic rhinitis hay fever, inflammation of the nasal pathways Alopecia hair loss 

Amenorrhoea absence of menstruation Angina thoracic pain due to lack of oxygen supply to the myocardium 

Candidiasis infection caused by Candida species Cardiotoxicity toxic effect to the cardiac tissues Cataract loss of transparency of the lens of the eye Chilblains areas of the skin that are inflamed and present as bluish-red in colour 

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From tlie definition of conditional probability in Eq. (19.5.1), one can deduce tlie logically equivalent definition diat event A and event B are independent if and only if... 

A very useful relationship, often called Bayes rule, exists between the two conditional probabilities P n in An in Bm and P m in Bm conditional probability implies the relationships,... 

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

Let the event b be the set of causes of an event a. Now from the definition of conditional probability the probability that bt occurs and a occurs is... 

Slurry reactors. For three-phase systems the definition of conditions at which (catalyst) particles are in motion is important. Two limiting states with respect to particle behaviour can be distinguished (1) complete suspension, i.e. all particles just move, and (2) uniform suspension, i.e. the particles are evenly distributed over the whole reaction zone. The power required to reach the second state is much higher, while uniform suspension is not often necessary. Circulation of the liquid with the dissolved gas is usually sufficiently fast to provide reactants to the surface of catalyst particles if they are suspended at all. 

The value of cultured stem cells to toxicology is in their promise as a continuous source of cells that can be induced to express a chosen cell-type-specific mature phenotype. This achievement would address two current limitations in safety testing scarcity of metabolically competent human systems and use of experimental animals. Numerous culture additives and conditions have been identified that cause commitment and partial differentiation along specific lineages, such as retinoic-acid-induced neuronal commitment of mouse embryonic stem cells. However, definition of conditions that produce fully differentiated cells with cell-type-specific function quantitatively similar to that of intact tissue remains problematic and is a high-priority research area in toxicology. 

Applying tlie definition of conditional probability Eq. (19.5.1) yields... 

Proof. With the definition of conditional entropy and Jensen s inequality,... 

With the definition of conditional mutual information and Rule (4 ), this yields H(5,IP = l(Si-,Histi i PK) + H(Si PK,Histi i) > and therefore... 

This finishes the proof of (3). For i = N+l, and with the-definition of conditional mutual information, one obtains... 

An agreed-upon condition on the completion date is an important project decision that should be made prior to award of construction contracts and should be made a written part of them. In the past, this was called mechanical completion, which was generally meant that all equipment was tested and ran in a mechanically approved manner and proper tests had been conducted to confirm tightness and pressure rating of system. Only necessary material for testing the system would be introduced. Today the requirements for cleanliness and proofing of tests as part of validation require that the definition of condition for turnover to start-up must be developed in much more detail. The questions to be asked when developing the completion plan are ... 

Now, by the definition of conditional probability, the probability that contrast n is significant, given that the previous n — 1 contrasts are, is the joint probability that contrast n is significant and the previous n — 1 contrasts are significant, divided by the probability that the previous n — 1 contrasts are significant. But the joint probability in question is nothing less than the probability that all n contrasts are significant. Hence the conditional probability we require is... 

Tong, C., Williams, G.M. 1980. Definition of conditions for the detection of genotoxic chemicals in the adult rat-liver epithelial cell/hypoxanthine-guanine phosphoribosyl... 

A sample space is generally defined and all probabilities are calculated with respect to that sample space. In many cases, however, we ate in a position to update the sample space based on new information. For example, like the fourth example of Example 2.3, if we just consider the case that two outcomes from roUing a die twice are the same, the size of the sample space is reduced from 36 to 6. General definitions of conditional probability and independence are introduced. The Bayes theorem is also introduced, which is the basis of a statistical methodology called Bayesian statistics. 

The Reverend Thomas Bayes [1702-1761] was a British mathematician and Presbyterian minister. He is well known for his paper An essay towards solving a problem in the doctrine of chances [14], which was submitted by Richard Price two years after Bayes death. In this work, he interpreted probability of any event as the chance of the event expected upon its happening. There were ten propositions in his essay and Proposition 3,5 and 9 are particularly important. Proposition 3 stated that the probability of an event X conditional on another event Y is simply the ratio of the probability of both events to the probability of the event Y. This is the definition of conditional probability. In Proposition 5, he introduced the concept of conditional probability and showed that it can be expressed regardless of the order in which the events occur. Therefore, the concern in conditional probability and Bayes theorem is on correlation but not causality. The consequence of Proposition 3 and 5 is the Bayes theorem even though this was not what Bayes emphasized in his article. In Proposition 9, he used a billiard example to demonstrate his theory. The work was republished in modern notation by G. A. Barnard [13]. In 1774, Pierre-Simon Laplace extended the results by Bayes in his article Memoire sur la probabilite des causes par les evenements (in French). He treated probability as a tool for filling up the gap of knowledge. The Bayes theorem is one of the most frequently encountered eponyms in the literature of statistics. 

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

The influence of the presence of the Mu ion on the titration of Mi may also be taken into account by introducing an appropriate conditional constant. As usual in the definition of conditional constants, we begin with the recognition of the principal and parasitic reactions. Here we consider the parasitic reaction is that of the Mu ion with EDTA. If, in order to simplify, we suppose that Mi reacts only with EDTA to give MiY and that EDTA reacts only with Mi and Mu (it gives rise to no further reaction, such as, for example, a protonation), we can then write... 

In general, the conditional distributions for each block of parameters, given all the other parameters and the observed data may be very complicated. By the definition of conditional probability... 

Since the effective propensities K(x x, t) are in terms of P(y x t), we need to know the dynamic behavior of these conditional probabilities. We have from the definition of conditional probability that... 

For characterisation of structure in the quasi-equilibrium state the approximate condition d = 2.5 was accepted above (see Equations 6.37 and 6.38). The authors of papers [101-103] carried out a more precise definition of conditions at which the quasi-equilibrium state was reached and they compared estimations of the different modes of those conditions. 

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