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Extra resources for Probability and Induction

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6) are called the n-dimensional distributions of the random function. Finite-dimensional distributions satisfy the following consistency conditions: 1. If i1 , . . , in is a permutation of 1, . . ,θn (B1 , . . ,θin (Bi1 , . . , Bin ); 2. ,θn−1 ,θn (B1 , . . ,θn−1 (B1 , . . 3 Random Mappings 43 (b) Moment functions. To specify all of the ﬁnite-dimensional distribution functions is constructively impossible. Therefore other parameters of random functions are used particularly the moment functions.

Suppose that n experiments have been performed in which ξ takes the values ξ1 , ξ2 , . . , ξn . Consider the average value of the resulting observations 1 1 x1 ξ¯ = (ξ1 + . . + ξn ) = IA1 + x2 IA2 + . . n n m1 m2 x1 + x2 + . . + xr IAr = n n + mr xr = n r xk νn (Ak ) . k=1 Here mi is the number of occurrences of Ai in the n experiments and νn (Ai ) is the relative frequency of Ai . If we replace the relative frequencies on the right-hand side by probabilities, we obtain xk P{ξ = xk }. It is natural to view this as the stochastic average of the random variable.

Xn ) = µ({y : y 1 < x1 , . . , y n < xn }), then F is the distribution function for µ. 4 Construction of Probability Spaces (1) 47 (n) ∆h1 . . ∆hn F (x1 , . . , xn ) = µ([x1 , x1 + h1 [× . . × [xn , xn + hn [) . 1) Theorem. 1). Proof. Consider the sets in Rn that are representable as a ﬁnite union of halfopen intervals in Rn of the form [a1 , b1 [×[a2 , b2 [× . . × [an , bn [ (ai may be −∞ and bi may be ∞). These sets form an algebra A0 . Every set in A0 is expressible as the union of disjoint half-open intervals.