Probability Inequalities in Multivariate Distributions

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Can the bounds in the multivariate Chebyshev inequality be attained?

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Statistics and Probability Letters, 91, pp. A multivariate Tchebycheff inequality.


The Annals of Mathematical Statistics, 29, pp. On generalised Tchebycheff theorems in the mathematical theory of statistics, Biometrika,12 3—4 , pp.

Financial applications of the Mahalanobis distance, Applied Economics and Finance, 1 2 , pp. We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners in accordance with our Privacy Policy.

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Statistics in Transition New Series , 20 3 , Address Of Delivery :. Moreover, the final row and the final column give the marginal probability distribution for A and the marginal probability distribution for B respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution. If a coin displays "heads" then the associated random variable takes the value 1, and it takes the value 0 otherwise. All possible outcomes are. Since the coin flips are independent, the joint probability density function is the product of the marginals:.

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The multivariate normal distribution , which is a continuous distribution, is the most commonly encountered distribution in statistics. When there are specifically two random variables, this is the bivariate normal distribution, shown in the graph, with the possible values of the two variables plotted in two of the dimensions and the value of the density function for any pair of such values plotted in the third dimension.

The probability that the two variables together fall in any region of their two dimensions is given by the volume under the density function above that region.


This identity is known as the chain rule of probability. The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete. With one variable of each type we have. Either of these two decompositions can then be used to recover the joint cumulative distribution function:.


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The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables. While the number of independent random events grows, the related joint probability value decreases rapidly to zero, according to a negative exponential law. This means that acquiring any information about the value of one or more of the random variables leads to a conditional distribution of any other variable that is identical to its unconditional marginal distribution; thus no variable provides any information about any other variable.

Such conditional independence relations can be represented with a Bayesian network or copula functions. Named joint distributions that arise frequently in statistics include the multivariate normal distribution , the multivariate stable distribution , the multinomial distribution , the negative multinomial distribution , the multivariate hypergeometric distribution , and the elliptical distribution. From Wikipedia, the free encyclopedia.

Probability inequalities in multivariate distributions /

Probability distributions. Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf—Mandelbrot. Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S U Landau Laplace asymmetric Laplace logistic noncentral t normal Gaussian normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy—Widom variance-gamma Voigt.