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Vysochanskij–Petunin Inequality
In probability theory, the Vysochanskij– Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper bound for the probability that it lies further away. The sole restrictions on the distribution are that it be unimodal and have finite variance. (This implies that it is a continuous probability distribution except at the mode, which may have a non-zero probability.) Theorem Let X be a random variable with unimodal distribution, and \alpha\in \mathbb R. If we define \rho=\sqrt then for any r>0, :\begin \operatorname(, X-\alpha, \ge r)\le \begin \frac&r\ge \sqrt\rho \\ \frac-\frac&r\le \sqrt\rho \\ \end. \end Relation to Gauss's inequality Taking \alpha equal to a mode of X yields the first case of Gauss's inequality. Tightness of Bound Without loss of generality, assume \alpha=0 and \rho=1. * If r. * If 1\le r\le \sqrt, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mode (statistics)
The mode is the value that appears most often in a set of data values. If is a discrete random variable, the mode is the value (i.e, ) at which the probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled. Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions. The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points , , etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently. When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Three (statistics)
In statistical analysis, the rule of three states that if a certain event did not occur in a sample with subjects, the interval from 0 to 3/ is a 95% confidence interval for the rate of occurrences in the population. When is greater than 30, this is a good approximation of results from more sensitive tests. For example, a pain-relief drug is tested on 1500 human subjects, and no adverse event is recorded. From the rule of three, it can be concluded with 95% confidence that fewer than 1 person in 500 (or 3/1500) will experience an adverse event. By symmetry, for only successes, the 95% confidence interval is . The rule is useful in the interpretation of clinical trials generally, particularly in phase II and phase III where often there are limitations in duration or statistical power. The rule of three applies well beyond medical research, to any trial done times. If 300 parachutes are randomly tested and all open successfully, then it is concluded with 95% confiden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantelli's Inequality
In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for \lambda > 0, : \Pr(X-\mathbb ge\lambda) \le \frac, where :X is a real-valued random variable, :\Pr is the probability measure, :\mathbb /math> is the expected value of X, :\sigma^2 is the variance of X. Applying the Cantelli inequality to -X gives a bound on the lower tail, : \Pr(X-\mathbb le -\lambda) \le \frac. While the inequality is often attributed to Francesco Paolo Cantelli who published it in 1928, it originates in Chebyshev's work of 1874. When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. The Chebyshev inequality has "higher moments versions" and "vector versions", and so does the Cantelli inequality ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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