Stochastic processes in discrete time
... p, q ∈ (0, 1), p + q = 1, and fn := ni=1 εi where n = 1, 2, ... and the random (p) variables εi were defined in Definition 2.1.1. Consider the event A := {ω : # {n : fn (ω) = 0} = ∞} . In words, A is the event, that the path of the process (fn )∞ n=1 reaches 0 infinitely many often. We would like to ...
... p, q ∈ (0, 1), p + q = 1, and fn := ni=1 εi where n = 1, 2, ... and the random (p) variables εi were defined in Definition 2.1.1. Consider the event A := {ω : # {n : fn (ω) = 0} = ∞} . In words, A is the event, that the path of the process (fn )∞ n=1 reaches 0 infinitely many often. We would like to ...
Permutation and Combination, Probability
... end, but there is nothing like a beginning or an end in a circular permutation. Thus, in a circular permutation, we consider one object as fixed and the remaining objects are arranged as in case of linear arrangements. ...
... end, but there is nothing like a beginning or an end in a circular permutation. Thus, in a circular permutation, we consider one object as fixed and the remaining objects are arranged as in case of linear arrangements. ...
GDPaper20131014 - Leicester Research Archive
... pairwise comparisons; and this increases to 45 pairwise comparisons when 10 different treatment regimens are included. Presenting such large numbers of results can be challenging, especially when NMA is used to evaluate a number of different outcome measures within the area of interest. Two recent r ...
... pairwise comparisons; and this increases to 45 pairwise comparisons when 10 different treatment regimens are included. Presenting such large numbers of results can be challenging, especially when NMA is used to evaluate a number of different outcome measures within the area of interest. Two recent r ...
Probability - UCLA Statistics
... The sample wont exactly resemble the population, there will be some chance variation. This discrepancy is called "chance error due to sampling". ...
... The sample wont exactly resemble the population, there will be some chance variation. This discrepancy is called "chance error due to sampling". ...
Deriving a Probability Density Calculator (Functional Pearl)
... If the only question we ever ask about a distribution is “what is its expected value?”, then it would be adequate for the meaning of each expression to equal its expected value. Unfortunately, there are other questions we ask whose answers differ on expressions with the same expected value. For exam ...
... If the only question we ever ask about a distribution is “what is its expected value?”, then it would be adequate for the meaning of each expression to equal its expected value. Unfortunately, there are other questions we ask whose answers differ on expressions with the same expected value. For exam ...
Empirical Statistical Model To Estimate the Accuracy of Peptide
... in order to derive a list of identified peptides and their corresponding proteins. This task necessarily entails distinguishing correct peptide assignments from false identifications among database search results. In the case of small datasets, this can be achieved by researchers with expertise manu ...
... in order to derive a list of identified peptides and their corresponding proteins. This task necessarily entails distinguishing correct peptide assignments from false identifications among database search results. In the case of small datasets, this can be achieved by researchers with expertise manu ...
Statistics
Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or societal problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as ""all persons living in a country"" or ""every atom composing a crystal"". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments.When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.Two main statistical methodologies are used in data analysis: descriptive statistics, which summarizes data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draws conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and a synthetic data drawn from idealized model. An hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is falsely rejected giving a ""false positive"") and Type II errors (null hypothesis fails to be rejected and an actual difference between populations is missed giving a ""false negative""). Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis.Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other important types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. The presence of missing data and/or censoring may result in biased estimates and specific techniques have been developed to address these problems.Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more heavily from calculus and probability theory. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data.