By John Maynard Keynes

With this insightful exploration of the probabilistic connection among philosophy and the heritage of technological know-how, the well-known economist breathed new existence into reports of either disciplines. initially released in 1921, this crucial mathematical paintings represented an important contribution to the idea concerning the logical likelihood of propositions, and introduced the “logical-relationist” concept.

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**Example text**

It can be considered as a regularity property of the only get that i=1 expectation. See [dF] for a thorough discussion of this problem. In the following we introduce some of the most common discrete distributions. 2 Bernoulli Scheme A simple and useful model from which some discrete distributions can be derived is the Bernoulli scheme. It can be thought of as a potentially infinite sequence of trials, each of them with two possible outcomes called success and failure. Each trial is performed in the same known conditions and we assume that there is no influence between different trials.

Possible cases = n A sample with i white balls contains (n − i) black balls. The number of favorable cases that correspond to such samples is therefore given by: f avorable cases = H i N−H n−i . The random number X is said to have hypergeometric distribution with parameters n, H, N . By the former discussion we have: 32 2 Discrete Distributions P(X = i) = N−H n−i H i N n . In order to compute the expectation of X , it is convenient to decompose it as n X= Ei i=1 where E i is the event that a white ball is chosen at ith drawing.

Er(i) for i ≥ 1. We assume that H1 , . . , Hn are stochastically independent for every n and that P(E k(i) ) = pk , k = 1, . . , r , for all i ≥ 1, with p1 + · · · + pr = 1. The sequence H1 , H2 , . . is called a generalized Bernoulli scheme. In the case r = 2 a generalized Bernoulli scheme is equivalent to the ordinary Bernoulli scheme (F1 , F2 , . ) where Fi = E 1(i) with parameter p = p1 . We can represent a generalized Bernoulli scheme in an array: E 1(1) , . . , Er(1) E 1(2) , . . , Er(2) .