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Mean-preserving Spread
In probability and statistics, a mean-preserving spread (MPS) is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function or probability mass function while leaving the mean (the expected value) unchanged. As such, the concept of mean-preserving spreads provides a stochastic ordering of equal-mean gambles (probability distributions) according to their degree of risk; this ordering is partial, meaning that of two equal-mean gambles, it is not necessarily true that either is a mean-preserving spread of the other. Distribution A is said to be a mean-preserving contraction of B if B is a mean-preserving spread of A. Ranking gambles by mean-preserving spreads is a special case of ranking gambles by second-order stochastic dominance – namely, the special case of equal means: If B is a mean-preserving spread of A, then A is second-order stochastically dominant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraposition
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The contrapositive of a statement has its Antecedent (logic), antecedent and consequent Inverse (logic), inverted and Conversion (logic), flipped. Material conditional, Conditional statement P \rightarrow Q. In Logical connective, formulas: the contrapositive of P \rightarrow Q is \neg Q \rightarrow \neg P . If ''P'', Then ''Q''. — If not ''Q'', Then not ''P''. ''"''If ''it is raining,'' then ''I wear my coat" —'' "If ''I don't wear my coat,'' then ''it isn't raining."'' The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true. The contrapositive ( \neg Q \rightarrow \neg P ) can be compared with three other statements: ;Inverse (logic), Inversion (the inverse), \neg P \rightarrow \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Google Books
Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical character recognition (OCR), and stored in its digital database.The basic Google book link is found at: https://books.google.com/ . The "advanced" interface allowing more specific searches is found at: https://books.google.com/advanced_book_search Books are provided either by publishers and authors through the Google Books Partner Program, or by Google's library partners through the Library Project. Additionally, Google has partnered with a number of magazine publishers to digitize their archives. The Publisher Program was first known as Google Print when it was introduced at the Frankfurt Book Fair in October 2004. The Google Books Library Project, which scans works in the collections of library partners and adds them to the digital invent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family of probability distributions is such that there is a parameter ''s'' (and other parameters ''θ'') for which the cumulative distribution function satisfies :F(x;s,\theta) = F(x/s;1,\theta), \! then ''s'' is called a scale parameter, since its value determines the " scale" or statistical dispersion of the probability distribution. If ''s'' is large, then the distribution will be more spread out; if ''s'' is small then it will be more concentrated. If the probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies :f_s(x) = f(x/s)/s, \! where ''f'' is the density of a standardized version of the density, i.e. f(x) \equiv f_(x). An estimator of a scale p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk (statistics)
Statistical risk is a quantification of a situation's risk using statistical methods. These methods can be used to estimate a probability distribution for the outcome of a specific variable, or at least one or more key parameters of that distribution, and from that estimated distribution a risk function can be used to obtain a single non-negative number representing a particular conception of the risk of the situation. Statistical risk is taken account of in a variety of contexts including finance and economics, and there are many risk functions that can be used depending on the context. One measure of the statistical risk of a continuous variable, such as the return on an investment, is simply the estimated variance of the variable, or equivalently the square root of the variance, called the standard deviation. Another measure in finance, one which views upside risk as unimportant compared to downside risk, is the downside beta. In the context of a binary variable, a simple s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Ordering
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable A may be neither stochastically greater than, less than nor equal to another random variable B. Many different orders exist, which have different applications. Usual stochastic order A real random variable A is less than a random variable B in the "usual stochastic order" if :\Pr(A>x) \le \Pr(B>x)\textx \in (-\infty,\infty), where \Pr(\cdot) denotes the probability of an event. This is sometimes denoted A \preceq B or A \le_ B. If additionally \Pr(A>x) x) for some x, then A is stochastically strictly less than B, sometimes denoted A \prec B. In decision theory, under this circumstance ''B'' is said to be first-order stochastically dominant over ''A''. Characterizations The following rules describe situations when one random variable is stochastically less than or equal to another. Stri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Utility Hypothesis
The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and Preference, preferences. The expected utility hypothesis states an agent chooses between risky prospects by comparing expected utility values (i.e. the weighted sum of adding the respective utility values of payoffs multiplied by their probabilities). The summarised formula for expected utility is U(p)=\sum u(x_k)p_k where p_k is the probability that outcome indexed by k with payoff x_k is realized, and function ''u'' expresses the utility of each respective payoff. On a graph, the curvature of u will explain the agent's risk attitude. For example, if an agent derives 0 utils from 0 apples, 2 utils from one apple, and 3 utils from two apples, their expected utility for a 50–50 gamble between zero apples and two is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Review Of Economic Studies
''The Review of Economic Studies'' (also known as ''REStud'') is a quarterly peer-reviewed academic journal covering economics. It was established in 1933 by a group of economists based in Britain and the United States. The original editorial team consisted of Abba P. Lerner, Paul Sweezy, and Ursula Kathleen Hicks. It is published by Oxford University Press. The journal is widely considered one of the top 5 journals in economics. It is managed by the editorial board currently chaired by Nicola Fuchs-Schündeln (Goethe University Frankfurt). The current joint managing editors are Thomas Chaney (Sciences Po), Andrea Galeotti (London Business School), Nicola Gennaioli (Bocconi University), Veronica Guerrieri (University of Chicago), Kurt Mitman (Institute for International Economic Studies, Stockholm University), Francesca Molinari (Cornell University), Uta Schönberg (University College London), and Adam Szeidl (Central European University). According to the ''Journal Citation Repor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Dominance
Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering. The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers. It is based on shared preferences regarding sets of possible outcomes and their associated probabilities. Only limited knowledge of preferences is required for determining dominance. Risk aversion is a factor only in second order stochastic dominance. Stochastic dominance does not give a total order, but rather only a partial order: for some pairs of gambles, neither one stochastically dominates the other, since different members of the broad class of decision-makers will differ regarding which gamble is preferable without them generally being considered to be equally attractive. Throughout the article, \rho, \nu stand for probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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