Probability, statistical term

Significance A statistical term relating to tests made to ascertain the probability of an effect or correlation. 

Taking a value of 107 for N would mean that in our galaxy (with its perhaps 100 billion stars), there could be several million planets with life forms capable of interstellar communication. However, if these were distributed statistically, the nearest would still be 200 light years away from Earth. One point is important the term probability used in the Drake equation is interpreted in the sense of subjective probability (a term from the nomenclature used by statisticians and probability theorists), as the numerical value of this probability is determined only by the experience of the scientist concerned (Casti, 1989). Casti also provides more information on the Drake factors (apart from the factor fs) in the chapter Where are they then In summary, we can say that the Drake equation is a first attempt to quantify the ETI problem in order to move from the area of science fiction and pure speculation to that of serious scientific debate. 

ISO 3534-1 2006 Statistics - Vocabulary and symbols -Part 1 General statistical terms and terms used in probability ... 

Maximum (maximized) likelihood is a statistical term that refers to the probability of randomly drawing a particular sample from a population, maximized over the possible values of the population parameters. Selected entries from Methods in Enzymology [vol, page(s)] Theory, 210, 203 testing by simulations, 210, 225 computer applications for, 210, 233 fitting of sums of exponentials to dwell-time distributions, 207, 772 fluorescence data analysis, 210,... 

A statistical term for the deviation from the true value within which lies an experimentally measured value with a probability of 0.50. This corresponds to 0.674 cr (i.e., 0.674 times the standard deviation). See Statistics (A Primer) Normal Error Curve... 

In order to take into account the spontaneity and irreversibility of real processes (heat always goes from a hot substance to a cold one, but not the reverse), thermodynamics invokes the notion of the entropy S. In statistical terms entropy is defined as the probability of accessible states for each molecule in the system ... 

ISO 3534-1 1993. Statistics - Vocabulary and Symbols - Part 1 Probability and General Statistical Terms, International Organization for Standardization, Geneva, 1993. 

Entropic factors are a major problem for relatively large molecules. For organic macromolecules, the simulation of the probability W(S=k-In (W)) by molecular dynamics calculations or Monte Carlo simulations, has been used to calculate the entropy from fluctuations of the internal coordinates189"921. For simple coordination compounds the corrections based on calculated entropy differences are often negligible in comparison with the accuracy of the calculated enthalpies116,63,881. Therefore, the relatively easily available statistical term (Sstat) is usually the only one that is included in the computation of conformational equilibria (see Chapters 7 and 8). 

Based on the sample data, we may reject the null hypothesis when in fact it is true, and consequently accept the alternative hypothesis. By failing to recognize a true state and rejecting it in favor of a false state, we will make a decision error called a false rejection decision error. It is also called a false positive error, or in statistical terms, Type I decision error. The measure of the size of this error or the probability is named alpha (a). The probability of making a correct decision (accepting the null hypothesis when it is true) is then equal to 1—a. For environmental projects, a is usually selected in the range of 0.05-0.20. 

Throughout this book, the approach taken to hypothesis testing and statistical analysis has been a frequentist approach. The name frequentist reflects its derivation from the definition of probability in terms of frequencies of outcomes. While this approach is likely the majority approach at this time, it should be noted here that it is not the only approach. One alternative method of statistical inference is the Bayesian approach, named for Thomas Bayes work in the area of probability. 

The sample of individuals is assumed to represent the patient population at large, sharing the same pathophysiological and pharmacokinetic-dynamic parameter distributions. The individual parameter 0 is assumed to arise from some multivariate probability distribution 0 / (T), where jk is the vector of so-called hyperparameters or population characteristics. In the mixed-effects formulation, the collection of jk is composed of population typical values (generally the mean vector) and of population variability values (generally the variance-covariance matrix). Mean and variance characterize the location and dispersion of the probability distribution of 0 in statistical terms. 

Even though the smallest MOEs in Figures 8.5-8.7 are relatively large, their true values are probably even larger. In statistical terms, this probable underestimation of the smaller MOEs occurs because the variance of an average of several variable events is less than the variance when every event is assnmed to have the same value. Thus, the lower percentiles in the distribution for an average of several events are larger than the lower percentiles in the distribution when every event is assumed to be the same. (For the same reason, the upper... 

International Organization for Standardization (ISO). Statistics— Vocabulary and symbols— Part 1 Probability and general statistical terms. ISO 3534-1, P ed. Geneva ISO, 1993. 

Statistics - Vocabulary and Symbols — Part 1 Probability and Genera Statistical Terms, (Revision of ISO 3534 1977), International Organisation for Standardisation, Geneva, Switzerland (1993). 

Project prioritization is a case where Bayesian statistics seems particularly appropriate. Every project is different and a naive frequentist view, defining probabilities in terms of limiting relative frequency of repeated events, seems both inappropriate and difficult. (But see below.) There is a considerable literature on eliciting subjective probabilities. 

Prior probability. In Bayesian statistics, a subjective probability assigned to a hypothesis (or statement or prediction) before seeing evidence. This is then updated after evidence is obtained, using Bayes theorem, in order to obtain a further subjective probability known as a posterior probability. The terms prior and posterior are relative to a given set of evidence. Once a posterior probability has been calculated it becomes available as a prior probability to be used in connection with future evidence. 

Notice that here we use the concept of distribution in a non-rigorous statistical sense. In rigorous statistical terms distribution usually alludes to the cumulative distribution function. Here, as in common language, by distribution we mean what in rigorous statistical terms is denoted as density function or probability function . 

Suppose a chemist synthesizes an analytical reagent that is believed to be entirely new. The compound is studied using a spectrometric method and gives a value of 104 (normally, most of our results will be cited in carefully chosen units, but in this hypothetical example purely arbitrary units can be used). From suitable reference books, the chemist finds that no compoimd previously discovered has yielded a value of more than 100 when studied by the same method under the same experimental conditions. The question thus naturally arises, has our chemist really discovered a new compound The answer to this question evidently lies in the degree of reliance that we can place on that experimental value of 104. What errors are associated with it If further study indicates that the result is correct to within 2 (arbitrary) units, i.e. the true value probably lies in the range 104 2, then a new material has probably been discovered. If, however, investigations show that the error may amount to 10 units (i.e. 104 10), then it is quite likely that the true value is actually less than 100, in which case a new discovery is far from certain. So a knowledge of the experimental errors is crucial (in this case as in every other) to the proper interpretation of the results. In statistical terms this example would involve the comparison of the experimental result (104) with a reference value (100) this topic is studied in detail in Chapter 3. 

A detailed understanding of all statistical terms and the formulas that are associated with their use is not an essential prerequisite to the practice of basic system safety analysis. A familiarization with their meaning is more than adequate for this purpose. The primary difference between statistics and probability is that probability attempts to predict the occurrence of future events, whereas statistics is used to develop models... 

Now I enlarge my coin base to two coins. What is the probabiUty of complete failure Where previously the probability of complete failure was one out of two possibilities (0.5 probability of eomplete failure), with two coins, there are four possibilities, out of which one represents two failures or complete failure (0.25 probability of such complete, or in statistical terms, conjoint failure). With a third coin, there is one way out of eight potential unique coin arrangements that we can get complete failure (probability of 0.125). Note that the toss of one eoin does not impact the alternatives for the next coin that is tossed, so the tosses are independent. Most people have an intuitive sense that with more independent events, the probability of everything going the same way beeomes relatively unlikely, and of course formal study of statistics serves to confirm this. 

---