Mathematical Ideas - Norfolk State University
... © 2008 Pearson Addison-Wesley. All rights reserved ...
... © 2008 Pearson Addison-Wesley. All rights reserved ...
Random projection trees for vector quantization
... for a comprehensive overview. In the most basic setup, there is some distribution P over RD from which random vectors are generated and observed, and the goal is to pick a finite codebook C ⊂ RD and an encoding function α : RD → C such that x ≈ α(x) for typical vectors x. The quantization error is u ...
... for a comprehensive overview. In the most basic setup, there is some distribution P over RD from which random vectors are generated and observed, and the goal is to pick a finite codebook C ⊂ RD and an encoding function α : RD → C such that x ≈ α(x) for typical vectors x. The quantization error is u ...
5. Conditional Expectation A. Definition of conditional expectation
... (b) (Monotone convergence) If {Xn } is an almost surely increasing sequence of random variables, meaning Xn+1 ≥ Xn , a.s. for each n, and X = a.s.- lim Xn is integrable, then limn→∞ E[Xn /G] = E[X/G], a.s. (c) (Fatou’s lemma) If {Xn } is a sequence of non-negative random variables, then E[lim inf Xn ...
... (b) (Monotone convergence) If {Xn } is an almost surely increasing sequence of random variables, meaning Xn+1 ≥ Xn , a.s. for each n, and X = a.s.- lim Xn is integrable, then limn→∞ E[Xn /G] = E[X/G], a.s. (c) (Fatou’s lemma) If {Xn } is a sequence of non-negative random variables, then E[lim inf Xn ...
Week 1: Simple probability.
... Alternatively, we can also assign probabilities based on our knowledge of physical properties, rather than from conducting numerous experiments. For example, if a coin is fair, then a natural model would be P(Heads) = 1/2 P(Tails) = 1/2. In this way, each outcome is equally likely. ...
... Alternatively, we can also assign probabilities based on our knowledge of physical properties, rather than from conducting numerous experiments. For example, if a coin is fair, then a natural model would be P(Heads) = 1/2 P(Tails) = 1/2. In this way, each outcome is equally likely. ...
Chapter 7 Random Variables
... are made to be identical in size, the width actually varies slightly from cup to cup. In fact, the diameter of a randomly-selected large drink cup is a random variable (C) that has a particular probability distribution. Likewise, the diameter of a randomly-selected large lid (L) is a random variable ...
... are made to be identical in size, the width actually varies slightly from cup to cup. In fact, the diameter of a randomly-selected large drink cup is a random variable (C) that has a particular probability distribution. Likewise, the diameter of a randomly-selected large lid (L) is a random variable ...
CHAPTER 2: Probability Sample Space: 2.1 A random experiment is
... the population will contract disease I sometime during their lifetime, 15% will contract disease II eventually, and 3% will contract both diseases. Find the probability that a randomly chosen person from this population will contract at least one disease. ...
... the population will contract disease I sometime during their lifetime, 15% will contract disease II eventually, and 3% will contract both diseases. Find the probability that a randomly chosen person from this population will contract at least one disease. ...
8.8 Probability
... event. With an uncertain event we are usually interested in which outcomes, if any, are more likely to occur than others, and to how large an extent. In tossing a coin three times, is it more likely that two heads or that one head will result? To answer such questions, we need a way to quantify the ...
... event. With an uncertain event we are usually interested in which outcomes, if any, are more likely to occur than others, and to how large an extent. In tossing a coin three times, is it more likely that two heads or that one head will result? To answer such questions, we need a way to quantify the ...
Model Estimation with Inequality Constraints: A Bayesian Approach Using SAS/IMLSoftware
... Economic models derived from economic . theory are generally acompanied with various sets of equality and inequality restrictions. For example, in an economic relationship between consumption,c, and income, y , c. = a + Pv. + et' the coefficient p represents the marginal propensity to consume and sh ...
... Economic models derived from economic . theory are generally acompanied with various sets of equality and inequality restrictions. For example, in an economic relationship between consumption,c, and income, y , c. = a + Pv. + et' the coefficient p represents the marginal propensity to consume and sh ...
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.