A Praktical Guide of Swing Trading Book by Swing L

By Swing L

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Let N be a nonnegative integer-valued random variable, independent of the Xns, with generating function N. Consider the sum S = Xl + ... + X N • Then s(t) = Et S = E[E(tSI N)] = E~(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.

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