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Extra resources for A Praktical Guide of Swing Trading Book
Let N be a nonnegative integer-valued random variable, independent of the Xns, with generating function
We refer to F. as the singular part of F. The discrete parts of F and F. coincide since F and F. have the same jumps everywhere. The remainder F. - Fd must therefore be singular continuous. We thus conclude that: Every df. can be written uniquely as a convex combination of a discrete, a singular continuous and an absolutely continuous df. s can be singular if and only if they are identical. A classic example of a singular continuous dJ. is the Cantor function. It is defined as follows. If k is the natural number with binary expansion k = L sj2j (Sj j = 0 or 1) denote Then the Cantor function is defined on [0,1] by _ 2k + 1 F (X ) - ~ l" Jor x (m = 1,2, ...
That is, we would like to find that (Borel) function g for which g(X) is as "close" to Y as possible. In many settings this "best approximation" works out to be the conditional expectation E(YIX). Definition I. Let X and Y be joint random variables and suppose Y has finite expectation. The conditional expectation E(YIX) is the random variable g(X), where g is defined by g(x) = E(YIX = x). (9) Thus the function g, which "best fits" X to Y is defined to be the conditional expectation of Y given that X = x, at each argument x.