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**Extra resources for A Praktical Guide of Swing Trading Book**

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Let N be a nonnegative integer-valued random variable, independent of the Xns, with generating function *~(t) = N(*

*X{t)). Similarly ES = E[E(SIN)] = E(NEX) = ENEX, 36 II. Multivariate Random Variables and by (11) + Var E(SIN) E(N Var X) + Var(NEX) = EN Var X + (EX)2 Var N. Var S = E Var(SIN) = This example will be useful in our discussion of branching chains. Orthogonal Projections We describe here an alternative approach to conditioning, useful in the mean square setting.*

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.