By Pierre Bremaud

ISBN-10: 0387964606

ISBN-13: 9780387964607

ISBN-10: 1461210461

ISBN-13: 9781461210467

Introduction to the elemental thoughts of chance conception: independence, expectation, convergence in legislation and almost-sure convergence. brief expositions of extra complicated themes equivalent to Markov Chains, Stochastic strategies, Bayesian choice conception and data Theory.

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**Example text**

To construct an injection f: E -> F, one must first select f(l) in F. There are n possibilities. Now, once f( 1) is chosen, there are only n - 1 possible choices for f(2) sinee f(2) must differ from f(I), etc. , in F - U(I), ... ,f(p - In. In summary, we have n(n - 1) ... (n - p + 1) possibilities for f. D Permutations. A special case of interest occurs when n = p. In this case, from Eq. (36), we see that A~ = n!. Now, if card E = card F, an injection of E into F is necessarily a bijection. Recalling that by definition a permutation of E is a bijection of E into itself, we see by specializing to the case E == F, that the number of permutations of a set with n elements is (37) Counting Subsets of a Given Size.

Xn(w))/n. Thio i~ thf' pmnirical frequency of heads in n tosses. ° 30 1. Basic Concepts and Elementary Models It is "known from experience" that Zn(w) tends to t as n goes to 00. 001111111111. that is to say w = t, and of other w's of a pathological kind for which Zn(w) does not tend to t as n -+ 00. The claim of Probabilists is that such w's are indeed pathological in the sense that 1 P({wilimZn(w) = t}) = I I· (46) This is the famous strong law of large numbers for heads and tails. It is a physical law in some sense, but here, in the mathematical setting of Probability Theory, it becomes a theorem, and it was proved by Borel in 1909.

Formula (39) is a particular case of the binomial formula (x + y)" = " (n)P I xPy"-P (40) (X, Y E IR). p=O It suffices to let X = Y = 1 in Eq. (40) to obtain Eq. (39). i 0( n) be real numbers. The product Xi 1 Xi 2 ... Xi p YJ' i ••• YJ' where n h, ... ,i p} is a subset of {I, ... ,n}, and Ul, ... ,j"-p} is the complement of {ij, ... ,ip} in {I, ... ,n}. Therefore, PROOF OF EQ. (40). Let Xi' Yi (I 0( [1 7=1 (Xj + yJ is formed of all possible products n [1 ;=t (Xi + y;) " = I ,. I p=O p I (ll •...

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