Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr(S-ES> nt) depend only on the endpoints of the ranges

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on Function The cumulative distribution function F X (x) for a random variable X(.), or when convenient represented by just X,is defined for all x as follows F X (x)=Pr{.:X(.)#x}Pr{X#x} It is sufficient information to calculate the probabilities of all allowable events and as a result is called a

We consider each row as a sample of random variables X1, X2,..., X5. In the cited paper N = 252 samples are given. H.-P. Helfrich (University of Bonn) Bayes ’ Statistics Brinkmann School 3 / 19Discrete example Counts Fat / circ. [60,80] (80,100] (100,120] (120,140] (140,160] [0,10] 18 18 0 0 0 (10

We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.

pure states mixed state possible states probabilities;

cardinality bounds on the auxiliary

Abstract. Random forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest. The generalization error for forests converges a.s. to a limit as the number of trees

• Random variables with a continuous range of possible values are quite common – The velocity of a vehicle

This paper deals with an expectation and a real-valued variance of fuzzy random variables. The expectation and the variance of a fuzzy random variable is characterized by Fr'echet's principle in a metric space. We study properties of the variance of a fuzzy random variables and compare