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K −1 and some Borel set F of . 3(a) has this form with k = 2, i = 0, and F = (1, 3). 3. PROBABILITY SPACES 41 1 and 3 in the x0 direction and between −∞ and ∞ in the x1 direction. To summarize, we have defined a space A with event space F, and an index set I such as Z+ , Z, , or [0,1), and we have formed the product space AI and the associated product event space F I . We know that this event space contains all one-dimensional events by construction. We next consider what other events must be in F I by virtue of its being an event space.

The generation of uniform random numbers is both a science and an art. Most function roughly as follows. One begins with floating point number in (0, 1) called the seed, say a, and uses another positive floating point number, say b, as a multiplier. A sequence xn is then generated recursively as x0 = a and xn = b × xn−1 mod (1) for n = 1, 2, . , that is, the fractional part of b × xn−1 . If the two numbers a and b are suitably chosen then xn should appear to be uniform. ) In fact, since there are only a finite number (albeit large) of possible numbers that can be represented on a digital computer, this algorithm must eventually repeat and hence xn must be a periodic sequence.

Which we denote by Z+ , and the space of all integers {. . , −2, −1, 0, 1, 2, . }, which we denote by Z. Other examples are the space of all rational numbers, the space of all even integers, and the space of all periodic sequences of integers. 3] are called discrete spaces. Spaces with finite or countably infinite numbers of elements are called discrete spaces. 4] An interval of the real line , for example, Ω = (a, b). We might consider an open interval (a, b), a closed interval [a, b], a half-open interval [a, b) or (a, b], or even the entire real line itself.

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