An Introduction to Probability Theory and Its Applications, Volume 2Wiley, 1950 - Probabilities Vol. 2 has series: Wiley series in probability and mathematical statistics. Bibliographical footnotes. "Some books on cagnate subjects": v. 2, p. 615-616. |
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Page 69
... normal density . Normal densities in higher dimensions will be introduced systematically in section 6. The excuse for anticipating the bivariate case is to provide an easy access to it . An obvious analogue to the normal density n of II ...
... normal density . Normal densities in higher dimensions will be introduced systematically in section 6. The excuse for anticipating the bivariate case is to provide an easy access to it . An obvious analogue to the normal density n of II ...
Page 84
... normal density then ( X1 , ... , Xn ) and ( Xn + 1 ) X , ) are independent iff Cov ( X ,, Xx ) = 0 for j ≤ n , k > n . Warning . The corollary depends on the joint density of ( X1 , X2 ) being normal and does not apply if it is only ...
... normal density then ( X1 , ... , Xn ) and ( Xn + 1 ) X , ) are independent iff Cov ( X ,, Xx ) = 0 for j ≤ n , k > n . Warning . The corollary depends on the joint density of ( X1 , X2 ) being normal and does not apply if it is only ...
Page 85
... normal density . Now E ( YY ) = 0 for k = 1 , ... , r - 1 and so Y , is independent of ( Y1 , ... , Y , -1 ) . But ( 6.8 ) 2 r - 1 ... rô2 = Y12 + · · · + Y2 - 1 + ( Y1 + ··· + Y , -1 ) 2 depends only on Y1 , ... , Y , -1 , and thus 2 ...
... normal density . Now E ( YY ) = 0 for k = 1 , ... , r - 1 and so Y , is independent of ( Y1 , ... , Y , -1 ) . But ( 6.8 ) 2 r - 1 ... rô2 = Y12 + · · · + Y2 - 1 + ( Y1 + ··· + Y , -1 ) 2 depends only on Y1 , ... , Y , -1 , and thus 2 ...
Contents
CHAPTER | 1 |
SPECIAL DENSITIES RANDOMIZATION | 44 |
PROBABILITY MEASURES AND SPACES | 101 |
Copyright | |
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An Introduction to Probability Theory and Its Applications, Volume 2 William Feller Limited preview - 1991 |
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a₁ applies arbitrary argument assume asymptotic atoms backward equation Baire functions Borel sets bounded central limit theorem characteristic function common distribution compound Poisson condition consider constant continuous function convergence convolution defined definition denote density derived distribution F distribution function equals example exists exponential distribution F{dx finite interval fixed follows formula given hence implies independent random variables inequality infinitely divisible integral integrand Laplace transform law of large left side lemma Let F limit distribution Markov martingale measure mutually independent normal distribution notation o-algebra obvious operator parameter Poisson process positive probabilistic probability distribution problem proof prove random walk renewal epochs renewal equation renewal process S₁ sample space satisfies semi-group sequence shows solution stable distributions stochastic stochastic kernel symmetric T₁ tends theory transition probabilities uniformly unique variance vector X₁ Y₁ zero expectation